Miniature inductors and related circuit component and methods of making same

ABSTRACT

New types of circuit elements for integrated circuits include structures wherein a thickness dimension is much greater than a width dimension and is more closely spaced than the width dimension in order to attain a tight coupling condition. The structure is suitable to form inductors, capacitors, transmission lines and low impedance power distribution networks in integrated circuits. The width dimension is on the same order of magnitude as skin depth. Embodiments include a spiral winding disposed in a silicon substrate formed of a deep, narrow, conductor-covered spiral ridge separated by a narrow spiral trench. Other embodiments include a wide, thin conductor formed in or on a flexible insulative ribbon and wound with turns adjacent one another, or a conductor in or on a flexible insulative sheet folded into layers with windings adjacent one another Further, a method of manufacture includes directional etching of the deep, narrow spiral trench to form a winding in silicon.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a Divisional of U.S. application Ser. No.15/285,310, filed Oct. 4, 2016, the contents of which are incorporatedherein by reference in their entirety for all purposes.

BACKGROUND OF THE INVENTION Introduction

This invention relates to miniature circuit components for applicationin connection with integrated circuits and certain types of miniaturediscrete circuits and methods for manufacturing such devices. Theinvention is related to improvements upon and replacement ofconventional miniature spiral inductors for example. Other embodimentsinclude transmission lines, capacitors, and low impedance powerdistribution networks in integrated circuits. More particularly, designtechniques and structures are disclosed with mathematical proofs thatpermit realization of small, very high current, and high inductancecomponents with good quality factor Q in one embodiment and also inapplications where high current distribution problems are found inminiature structures such as integrated circuit chips and powersupplies.

A class of devices called spiral inductors is known for use withsemiconductor devices, and they represent an example of the types ofdevices that can be made according to the invention. Spiral inductorsare defined as planar structures having a characteristic inductance, andthey are known to some degree for realizing inductors in integratedcircuit and packaging technologies. They are extensively used in RFIC(Radio Frequency Integrated Circuits) in LNA (Low Noise Amplifier), VCO(Voltage Controlled Oscillators), PLL (Phase Locked Loops) designs.Recently, spiral inductors have been considered for DC/DC converterarchitectures, mainly for buck converter designs for supplying on-chip,high current, low voltage requirements for processor designs.[3-9][Reference numbers refer to listed reference items below.]

Accurate analysis in preparation for suitable designs requires a goodunderstanding of complex electromagnetics and suggests a need forsophisticated and accurate analysis software. Designing a spiralinductor with a desired inductance value and AC performance in asmallest possible area is a more difficult optimization problem thananalyzing a given structure. Extending the concept of a small areaspiral inductor for use in power space as in DC/DC converterarchitectures puts even more difficult constraints on the design for agiven process technology, due to the high current specifications (1A-130 A) and much larger needed inductance values (10 nH-80 nH) within areasonably small area, such as regions not exceeding several hundredmicrons on a side for applications operating below 200 MHz with goodquality (Q>10 (Quality Factor). This is a nearly impossible problem tosolve with current process technologies related to integrated circuitarchitectures [3-24]. Therefore, the only presently known option is useof off-chip inductors. Such inductors suffer from undesirable issues indesign [3-24] and are not inherently small relative to the dimensions ofelements in a semiconductor device, resulting in a large area penalty onthe overall system dimension.

The electromagnetics of any inductor is complex. Even a distributedcircuit model is highly complex compared to capacitor and resistors. Thedistributed circuit model requires many mutual coupled inductors,resistors and capacitors that need to be calculated from the physicalgeometry. However a distributed circuit model is still is a simpler wayof understanding the circuit behavior of a spiral inductor compared tocomplex electromagnetics associated with it. To understand the inventionexplained it is useful to start from inductor basics. In this work bothcomplex electromagnetic simulations and circuit models derived from thesimplified solution of the three-dimensional simulation results areused. Thus, this disclosure is a fundamental tutorial for realizingminiature inductance devices.

The inventor would like to acknowledge the assistance of Prof. Dr. YusufLeblebici and Dr. Seniz E. Kucuk Eroglu of the Microelectronic SystemsLaboratory Swiss Federal Institute of Technology, Lausanne, Switzerlandfor providing technical assistance in manufacturing structures as hereindisclosed and Michael Brunolli of San Diego, Calif., for introducingthis very challenging problem, circuit level discussions and hisconstant encouragements throughout the work.

REFERENCES

The following references, many of which are herein cited, providesupplemental and background information for this invention and areincorporated herein by reference for all purposes. No representation ismade as to the extent of relevance to the present invention.

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SUMMARY OF THE INVENTION

According to the invention, miniature circuit components of variousdisclosed manufacturing techniques are provided that are suitable forforming or embedding in semiconductor structures or coupling tominiature circuitry and suitable as inductors, capacitors, transmissionlines and power distribution networks based on a principle of tightcoupling among conductive segments. In a specific embodiment, aminiature inductor is provided for integrated circuit applications thatcomprises a rectangular-cross-section metal conductive element formed invarious configurations as a multiple-turn winding formed of adjacentsegments and disposed such that the thickness dimension is much greaterthan the width dimension and windings are closely spaced in the widthdimension on a scale comparable to the width dimension to attain tightcoupling across multiple turns. Further embodiments include a spiralwinding disposed in a silicon substrate formed of a deep, narrow,conductor-covered spiral ridge separated by a narrow spiral channeltrough or valley. Other embodiments, useful especially for experimentalverification, comprise wide, thin conductors (corresponding toconventional windings) that are formed on a thin, flexible insulativesubstrate sheet (Flex Technology) wherein the sheet is wound into a rollwith the conductor-formed windings adjacent one another. Alternativelyconductors are formed on thin insulative sheets and folded or stackedinto layers with the conductor-formed windings adjacent one another. Theconductors of the alternative-embodiment devices may be fabricated usingflex-film processing techniques. The conductors may be disposed in acontinuous spiral or a rectangular spiral configuration and attached toa conventional semiconductor chip via interposers. The conductors are ofsufficient surface area to support intended or design currents atintended or design frequencies in both cases substantially greater thanis heretofore known due to achievement of very tight coupling acrossmultiple windings. Further according to the invention, a method ofmanufacture of inductors comprises forming a narrow spiral ridge ofsubstantial depth by etching into an insulative substrate that istypically of silicon, then the substrate is electroplated with aconductor, and finally the valley between the ridges is directionallyetched to remove conductor to establish continuous, closely spacedwindings of a width substantially greater than conductor thickness orthe ridge depth. The manufacturing techniques are adaptable totransmission lines, and capacitors.

The disclosed design parameters allow realization of very small area,large value and high Q inductors when operated at lower frequency rangessuited to power applications, as well as realization of relativelyhigh-Q, small-area inductors at any frequency range for a given processtechnology. Sizes of such devices make them compatible with manysemiconductor chip designs. Inductors constructed according to theinvention can achieve inductance values heretofore not achievable inminiature inductors.

The invention will be better understood upon reference to the followingdetailed description in connection with the accompanying drawing andembedded tables. The written descriptions in the drawings and the tablesform an integral part of this disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 0.1 is a top view of a spiral winding inductor according to a firstembodiment of the invention.

FIG. 0.2 is a perspective view of a spiral winding inductor of FIG. 0.1.

FIG. 0.3 is a top view of a flex spiral winding inductor according to asecond embodiment of the invention.

FIG. 0.4 is a perspective view of the inductor of FIG. 0.3 .

FIG. 0.5 is a perspective view of a third embodiment of the inventionillustrating a flex-formed folded coil.

FIG. 1.1 is a diagram for illustrating parameters for analyzing aninductor.

FIG. 1.2 is a graph for illustrating how width and thickness of aconductive winding relate.

FIG. 1.3 is a first diagram for defining the relationship of mutualinductance.

FIG. 1.3 .1 is a diagram for defining length and spacing.

FIG. 1.3 .2 is a second diagram for defining the relationship of mutualinductance.

FIG. 1.4 is a graph showing the relationship of thickness tocross-sectional area of a conductive winding.

FIG. 1.5 .1 is a graph relating inductance to length of an inductor tocross-sectional area and thickness.

FIG. 1.6 .1 is a top view of a square spiral-type inductor according tothe invention to be built into an interposer structure illustratinganalysis parameters according to the invention.

FIG. 1.7 .1 is a cross-sectional side view of a first interposerstructure-built spiral inductor showing C4 bump terminals on top andbottom of the interposer for external connection and IC connection.

FIG. 1.7 .2 is a cross-sectional side view of a first coupling betweenan interposer structure and an integrated circuit.

FIG. 1.8 .1 is a Cross-Sectional Side View of a Second InterposerStructure-Built Spiral inductor with C4 bump terminals connected on oneside of the device.

FIG. 1.8 .2 is a Cross-Sectional Side View of a Second Coupling Betweenan Interposer structure (inverted as compared to FIG. 1.8 .1) and anintegrated circuit.

FIG. 1.9 .1 is a cross-sectional side view with parameters for analyzingan interposer structure inductor.

FIG. 1.9 .2 is a cross-sectional side view with parameters for analyzinga further interposer structure inductor in accordance with theinvention.

FIG. 1.10 is a graph illustrating skin depth as a function of frequency.

FIG. 1.11 .1 is a graph illustrating the Bernoulli Function.

FIG. 1.11 .2 is a graph illustrating a detail of the Bernoulli Function.

FIG. 1.12 is a diagram in side cross-sectional view of a multi-layerinductor with parameters marked as used for analysis.

FIG. 1.13 .1 is a graph illustrating a solution for optimal thickness inan inductor according to the invention.

FIG. 1.14 .1 is a graph illustrating a range of solutions for optimalthickness in an inductor according to the invention.

FIG. 1.14 .2 is a graph illustrating a detail of FIG. 1.14 .1.

FIG. 1.15 .1 is a graph illustrating a range of solutions for optimalthickness.

FIG. 1.15 .2 is a graph illustrating a detail of FIG. 1.15 .1.

FIG. 1.16 .1 is a graph illustrating a range of solutions for optimalthickness.

FIG. 1.16 .2 is a graph illustrating a detail of FIG. 1.16 .1.

FIG. 1.17 .1 is a graph illustrating a range of solutions for optimalthickness.

FIG. 1.17 .2 is a graph illustrating a detail of FIG. 1.17 .1.

FIG. 1.18 .1 is a graph to show the magnitude of QPEAK variation as afunction of frequencies for various inductors 20, 40, 60 and 80 nH

FIG. 1.18 .2 is a graph to show the uniform current density and currentdensity distribution with “single sided solution” approximation.

FIG. 1.19 is a graph showing physically practical solutions at variousthicknesses and Q.

FIG. 1.20 .1 is a graph showing normalized magnetic fields across afour-winding inductor.

FIG. 1.20 .2 is a graph showing relative magnetic field in the innermostwinding.

FIG. 1.21 .1 is a graph illustrating a single sided solution vs. acomplete solution for various widths.

FIG. 1.21 .2 is a graph showing a detail of FIG. 1.21 .1.

FIG. 1.22 is a graph showing a relationship between Q and length.

FIG. 1.23 is a top plan view drawing of a spiral inductor constructedaccording to the invention.

FIG. 1.24 is a portion of a perspective view of a spiral inductoraccording to the invention.

FIG. 1.25 is side cross-sectional view of a spiral inductor according tothe invention with the mathematical matrix equivalent of the inductor.

FIG. 1.26 is a plot of a Table from Grover of Log(k) used in defining ageometric mean distance (g.m.d.) as a function of the ratio of w vs. dbetween two rectangles.

FIG. 1.27 is a plot of a further Table from Grover of Log(k) used indefining a geometric mean distance (g.m.d.) as a function of the ratiooft vs. d between two rectangles.

FIG. 1.28 is a plot of a further Table from Grover showing ranges ofLog(k) and Log(k) about very thin structures.

FIG. 1.29 is a graph illustrating g.m.d. for adjacent rectangles havingnarrowest sides facing one another.

FIG. 1.30 is a first combined graph to illustrate the differences in twocases of adjacent rectangles, where Case 2 is a preferable embodimentfor an inductor having a winding cross section according to theinvention.

FIG. 1.31 .1 is a second combined graph to illustrate the differences intwo cases of mutual inductance and self-inductance, where Case 2 is apreferable embodiment for the cross section and spacing of a winding ofan inductor according to the invention.

FIG. 1.31 .2 is a log scale graph of FIG. 1.31 .1.

FIG. 1.32 is a third combined graph to illustrate the differences in twocases, where Case 2 is a preferable embodiment of spacing of rectangularwindings of an inductor according to the invention as based on couplingcoefficients.

FIG. 1.33 is a schematic representation of windings to illustrate tightcoupling aspect ratio relations vs. adjacent spacing width and number ofturns.

FIG. 1.34 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio and spacing aspect ratio todetermine coupling to non-adjacent turns beyond 1.

FIG. 1.35 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio and spacing aspect ratio todetermine coupling to non-adjacent turns beyond 2.

FIG. 1.36 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio showing impact on total inductanceat 50μ metal thickness conditions for 500μ nominal length of four turnsaccording to the invention.

FIG. 1.37 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio showing impact on total inductanceunder 100μ metal thickness conditions for 500μ nominal length of fourturns according to the invention.

FIG. 1.38 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio showing impact on total inductanceunder 200μ metal thickness conditions for 500μ nominal length of fourturns according to the invention.

FIG. 1.39 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio showing impact on total inductanceunder 300μ metal thickness conditions for 500μ nominal length of fourturns according to the invention.

FIG. 1.40 is a graph illustrating the relation between number of tightlycoupled turns and metal aspect ratio showing impact on total inductanceunder 50, 100 and 200μ metal thickness conditions for a nominal lengthof 600μ according to the invention.

FIG. 1.41 is a graph illustrating Q as a function of metal aspect ratiofor 600μ nominal length of four turns according to the invention.

FIG. 1.42 is a graph illustrating inner to outer winding width afunction of metal aspect ratio for five turns allowing smallconstruction according to the invention.

FIG. 1.43 is a graph in log scale illustrating inner to outer windingwidth a function of metal aspect ratio for five turns allowing smallconstruction according to the invention.

FIG. 1.44 is a top plan view of a segment of an inductor according tothe invention.

FIG. 1.45 is a perspective view of a segment of an inductor according tothe invention.

FIG. 1.46 is a top plan view of two adjacent segments of an inductoraccording to the invention.

FIG. 2.1 through-FIG. 2.10 show first processing steps of the HARMSprocess according to the invention.

FIG. 2.11 is a SEM image of a grid of test structures illustratingsilicon core encapsulated by tantalum.

FIG. 2.12 is an expanded SEM image of a grid of test structuresillustrating silicon core encapsulated by tantalum.

FIG. 2.13 is a schematic of a spiral structure according to theinvention illustrating a preferred alignment of pad layout forinterconnection of an interposer-type structure.

FIG. 2.14 is a top plan view of a pad for an interposer structure.

FIG. 2.15 through FIG. 2.19 are illustrations of the result of processsteps for interconnection of interposer C4 bumps.

FIG. 2.20 is a schematic top plan view illustrating wafer scaleelectroplating of interposer structure devices according to theinvention.

FIG. 2.21 through FIG. 2.26 are illustrations of the result ofprocessing steps involved for backside C4 bumping for SiO wafers.

FIG. 2.26 .1 through FIG. 2.26 .4 are illustrations of the result ofprocessing steps involved in making fully integrated structuresaccording to the invention for buried oxide thicknesses between one andtwo microns and active IC layer between one and three microns, thuseliminating the interposer structure.

FIG. 2.26 .5 through FIG. 2.26 .11 are illustrations of the result ofprocessing steps involved in making fully integrated structuresaccording to the invention for buried oxide thicknesses between one andtwo microns and active IC layer greater than three microns, thuseliminating the interposer structure.

FIG. 2.26 .12 through FIG. 2.26 .16 are illustrations of the result ofprocessing steps involved in making fully integrated structuresaccording to the invention for a buried oxide thickness of about 20 nmaccording to the SIMOX process and an active layer of less than about200 nm, thus eliminating the interposer structure.

FIG. 2.27 is a first schematic side view of adjacent windingsillustrating critical dimensions of a first size.

FIG. 2.28 is a second schematic side view of adjacent windingsillustrating critical dimensions of a second size.

FIG. 2.28 .1 is a graph illustrating silicon core and total windingwidth for different silicon core thicknesses.

FIG. 2.28 .2 is a graph illustrating silicon to silicon core spacing fordifferent silicon core thicknesses.

FIG. 2.29 is a graph illustration needed Si Core thickness t_(3si), toachieve a desired Δ_(w) values for δ_(EM)=2, 4, 6 and 8μ.

FIG. 2.30 is a graph illustrating the cross-sectional area ofelectro-plated metal area that determines the resistance of a spiralinductor.

FIG. 2.31 .1 is a graph illustrating the amount of current capacity as afunction of silicon core thickness for different thicknesses ofelectroplating of a spiral inductor

FIG. 2.31 .2 is a graph illustrating the resistance vs. silicon corethickness of a 5 mm long line for different thicknesses ofelectroplating of a spiral inductor.

FIG. 2.32 is a schematic side cross-sectional view illustrating C4bumping on both sides of an interposer structure of a spiral inductoraccording to the invention.

FIG. 2.33 is a graph illustrating maximum allowable DC resistances forDC/DC converter inductors.

FIG. 2.34 .1 is a graph illustrating the relationship between corecolumn numbers and silicon and electroplating thickness in FIG. 2.17 .1.

FIG. 2.34 .2 is a graph illustrating the relationship between “siliconthrough via” (STV) resistance and inductor contact resistance vs.silicon core thickness and electroplating thickness.

FIG. 2.35 is a top pan view of an on-chip capacitor according to theinvention.

FIG. 3.1 is a diagram illustrating prior art standard parameters of aFlex PCB structure.

FIG. 3.2 is a diagram illustrating prior art standard dimensions of aFlex PCB structure.

FIG. 3.3 is a diagram of a prior art standard Flex to PCB connector.

FIG. 3.4 is a diagram of a test structure for three different widths ofFlex PCB traces as may be implemented according to the invention.

FIG. 3.5 is a schematic top plan view of a portion of a tested structurewith a standard connector.

FIG. 3.6 is a schematic top plan view of a Flex PCB inductor accordingto the invention showing Flex to PCB connector mounting.

FIG. 3.7 is a schematic side cross-sectional view of a Flex PCB inductoraccording to the invention showing Flex to PCB connector mounting.

FIG. 3.8 is a schematic side cross-sectional view of a prior arttwo-layer Flex PCB ribbon illustrating dimensions.

FIG. 3.9 .1 is a schematic top plan view of a two-port transformer.

FIG. 3.9 .2 is a schematic top plan view of a transformer with asecondary with center tap.

FIG. 3.10 is a schematic top plan view of a balun with voltage gain.

FIG. 3.11 is a schematic top plan view of a balun with voltage drop.

FIG. 4.1 is schematic top plan view of a folded stacked Flex structure.

FIG. 4.2 is a schematic top plan view for defining critical dimensionsof a folded stacked Flex structure.

FIG. 4.3 is a schematic top plan view of a folded stacked Flex structureillustrating that the segments between folds need not be straight.

DETAILED DESCRIPTION OF THE INVENTION

Referring to FIG. 0.1 , there is shown a top view of a spiral inductor10 embedded in a silicon substrate 11 according to a first embodiment ofthe invention, wherein the winding 12 is laid out in a rectangularpattern, width w 18 being comparable to the spacing s 16 betweensegments of the winding, and the depth or thickness t 14 of the winding12, not shown in FIG. 0.1 but as shown in the perspective view of FIG.0.2 , being substantially greater than the width w 14, in accordancewith the invention. The ratio of thickness to width may be on the orderof 5, 10, 15 or even 20 to 1, all within the contemplation of theinvention based on specific design criteria for inductance,current-carrying capacity and maximum operational frequency. Terminals20, 22 connect via interposers 24, 26. Dimensions of this structure foran 80 nH inductor are typically less than one mm across and in certainapplications are capable of inductance in excess of 100 nH whilesustaining high currents (in excess of several Amperes) and/or highfrequencies (in excess of 100 MHz) with good Q (in excess of 10), alldepending upon certain design considerations, such as windingcross-sectional area (w×t), inter-winding spacing, conductor thickness,conductor width and material resistivity, as well as winding length, ashereinafter defined and shown by way of example. The element 10 may befabricated in a silicon substrate and mounted on a silicon circuit (notshown) within a restricted pad area. (This disclosure does not teach howto form an inductor of this design in a semiconductor circuit, as thatis beyond the scope of this disclosure.)

FIG. 0.2 illustrates the device 10 of FIG. 0.1 showing that thethickness t 14 to be substantially greater than either width w 18 orspacing s 16. A novel manufacturing process is disclosed hereinafter.

Alternative designs, are shown in FIG. 0.3 , FIG. 0.4 and FIG. 0.5 . Ashereinafter explained, is fabricated of a conductive winding in a ribbonmanufactured according to Flex PCB Technology adapted to miniaturizedapplication. It is wound tightly around a central air core forming aweb. The core may be circular or rectangular. Accordingly the thicknessdimension is on a curved plane or series of flat planes and the widthdimensions of the winding are juxtaposed and spaced apart only byminimal space and the thickness of the ribbon.

Device 310 of FIG. 0.5 may likewise be formed according to Flex PCBtechnology. However, the winding is embedded in a ribbon laid out in aserpentine pattern and the ribbon is folded against itself in a zip-zagpattern so as to separate winding segments by the substrate. The windingsegments are arranged to be at least partially juxtaposed and separatedonly by the thickness of the substrate and any air gap.

Only a few windings are shown for illustrative purposes, but there arenormally four or more, up to about ten windings. Various patterns ofserpentine traces embedded in ribbon may be realized, differing fromlayer to layer, although a circular or rectangular spiral is likelysimpler to design, to fabricate, to analyze and to test. The embodimentof FIG. 0.5 may be mounted either horizontally (flat) on a semiconductorsubstrate, or vertically, with terminals A and B. The Flex PCBembodiments are specifically useful for circuit verification purposes.Due to the manufacturing processes, air gaps, winding tightness andalignment may be inconsistent, but the devices are useful as inexpensiveproof-of-concept and breadboard components.

More precise tolerances may be achieved where the devices aremanufactured in a semiconductor substrate, particularly as hereindisclosed. The steps are explained in more detail hereinbelow withrespect to actual structures.

The basic steps are as follows:

Step A, a semiconductor substrate of silicon over silicon oxide ofsuitable thickness is provided. If an interposer design is contemplated,then the substrate is stock material. If the inductor is to beintegrated into the same chip with a semiconductor circuit, the circuitis formed first and is usually in the silicon layer the surface of thechip or wafer opposite from the inductor.

Step B, a spiral channel 602 is etched into the silicon substrate thatis of sufficient width to leave a spiral ridge of width less thanwinding width w and of sufficient separation for a winding spacing s toyield a continuous ridge in a spiral surface pattern of a design length,wherein height of the ridge is sufficient to establish winding thicknesst. Thus the channel is of a depth corresponding to the design thicknesst and spacing s+w in the substrate to yield a continuous ridge in aspiral surface pattern (cf. FIG. 0.1 or FIG. 0.2 ) and of a designlength. The height of the ridge 604 is less than or equal to thickness tas hereinafter explained in connection with the design parameters.

Step C, a binding material 606 is applied to the surface of the ridgedsemiconductor material, for example titanium nitride, tantalum or likecommon semiconductor-to-metal adhesive. Application is by means ofconventional processing, and it should at least cover the length, widthand depth of the ridge 604.

Step D, a conductor 608, such as copper, aluminum or gold metal, isplated onto the entire ridge 604, including all tops and sidewalls, andbound to the ridge 604 by the binder. For this purpose, electroplatingis suitable, since it is capable of adhering to sidewalls. The build-updepth of the plating process is determined by the intended spacing sbetween plated facing walls of the ridge 604. The width of the ridgeplus the combined thickness of the plating corresponds to the intendedwidth of the winding w.

Step E, the bottom 610 of the channel 602 is etched away, thus formingan air gap between walls of the ridge 604 along its length and formingthe winding of thickness t, width w separated by spacing s betweenwindings. Both conductive faces of the ridge 602 are conductively coupleacross the top. This selective etching process is realized by thetechnique of directional etching 612. Directional etching includes forexample deep silicon reactive etching (the Bosch process), plasmaetching or possibly ion beam etching, along a spiral path tracing thebottom 620 of the channel 602. To the extent that side walls may beetched in the process, the electroplating step may include addingsufficient material to the side walls to compensate for residualetching.

Alternative etching techniques, such as use of photoresist patterns arealso within the contemplation of the invention. Moreover, the technologyto perform the semiconductor fabrication process may be an older,larger-spacing processing technology than is used for fabrication of theassociated semiconductor chip. It can now be seen how a high inductance,high current capability, high frequency, high Q, and closely spaced,high-aspect-ratio-winding inductor can be realized that is suitable forembedding on (and eventually in) a semiconductor chip.

The following is a design tutorial leading to an understanding of theparameters employed according to the present invention, plus adescription of specific embodiments of the invention made according tothese design criteria.

Self and Mutual Inductance Calculations

An excellent source of inductance and mutual inductance calculations canbe found in the classic book Inductance Calculations by FrederickGrover, first published in 1946 [1]. Due to demand by those working inthe field it has been re-printed many times and remains a valuablesource. In that work, numerical calculations of any arbitrarycross-sections for self inductance and mutual inductance of theirarbitrary arrangements in spatial coordinates are used. What followshere is a practical guide for design with respect to self-inductance andmutual inductance considerations.

The general formula of the DC self inductance (internal inductance foruniform current density distribution) of a prism geometry with any givencross-section can be given as the general inductance formula [1]:

$\begin{matrix}{L_{GENERAL} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{r} \right)} - 1 + \frac{\delta_{1}}{l}} \right\rbrack}}} & (1.1)\end{matrix}$

where l, r, δ₁ are in cm and (1.1) gives L_(GENERAL) in Henry units, rand δ₁ being geometric- and arithmetic-mean-distance-related quantities,which are related to the cross-sectional geometry of the prism, which isassumed to have a uniform current distribution flowing through itscross-section. Geometric Mean Distance (g.m.d) and Arithmetic MeanDistance (a.m.d) are important concepts in inductance and mutualinductance calculations [1,2]. The “geometric mean distance” (g.m.d.)between any pair of n points can be expressed as,

$\begin{matrix}{{g.m.d.} = {\left( {\prod\limits_{i = 1}^{n}r_{i}} \right)^{\frac{1}{n}} = {\exp\left\lbrack {\frac{1}{n}{\sum\limits_{i = 1}^{n}{\ln\left( r_{i} \right)}}} \right\rbrack}}} & (1.2)\end{matrix}$

Similarly, “arithmetic mean distance” (a.m.d.) can be expressed as,

$\begin{matrix}{{a.m.d.} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}d_{i}}}} & (1.3)\end{matrix}$

where r₁ and d_(i) are distances between pairs of points in selectedregions as can be shown in FIG. 1.1 , where P(i)_(u). and P(i)_(v)represent cross sections of two adjacent conductive structures throughwhich current may flow, commonly corresponding with the cross section ofwindings of an inductor, which are generally NOT rectangular in crosssection. As will be explained hereinafter, rectangular cross sectionsare an important design feature of the invention hereafter explained.The summations (1.2) and (1.3) for very large n can be generalized usingmultiple integrals performed in closed areas which have very interestingapplications in many partial differential equation problems encounteredin electromagnetics and quantum mechanics [2]. These integrals take onthe form of quadruple integrals, which also can be calculated bynumerical integration with some care in Cartesian coordinates.

The general inductance formula (1.1) can be written explicitly forspecific cross-sectional geometries. For rectangular cross-sections, ofthe type upon which this invention is focused, Equation (1.1) becomes,

$\begin{matrix}{L_{RECT} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack}}} & (1.4)\end{matrix}$ where0 ≤ f₁(w, t) ≤ 0.00249 l, w, t → cm → L_(RECT)[H]

where, l, w, and t are the length, width and thickness, respectively.

For a circular cross-section the general formula Equation (1.1) becomes,

$\begin{matrix}{L_{CIRC} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{r} \right)} - {{0.7}5}} \right\rbrack}}} & (1.5)\end{matrix}$ l, r → cm → L_(CIRC)[H]

where, r is the radius of the circular cross-section.

For a coaxial or ring cross-section which is also named as a “hollowcylinder” Equation (1.1) becomes,

$\begin{matrix}{L_{RING} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{r_{1}} \right)} + {f_{2}\left( {r_{1},r_{2}} \right)} - 1} \right\rbrack}}} & (1.6)\end{matrix}$ where0 ≤ f₂(r₁, r₂) ≤ 0.25 l, r₁, r₂ → cm → L_(RING)[H]

where, r₁ and r₂ are the outer and inner radiuses of the coax, ring orthe hollow cylinder.

For elliptical cross-section where a and b are the major and minor axesof the ellipse, (1.1) becomes,

$\begin{matrix}{L_{ELLIPS} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{a + b} \right)} - {{0.7}5}} \right\rbrack}}} & (1.7)\end{matrix}$ l, a, b → cm → L_(ELLIPS)[H]

A closer look at relations (1.4-1.7) shows that there is a minimumlength l_(MIN) as a function of the cross-sectional dimensions wherethese formulas apply. For the rectangular cross-section, which is themain focus in this invention, l_(MIN) can be expressed as a function ofw and t by solving,

$\begin{matrix}{{{\ln\left( \frac{2l_{MIN}}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} = 0} & (1.8)\end{matrix}$

The function ƒ₁(w,t) is plotted in FIG. 1.2 using the same notation asgiven in Grover [1]. Being an older work, the notation used is no longerconventional. The reason for using Grover's notation is to retain theexpressions as original as possible. As can be seen the maximum valuefor ƒ₁(w,t) is 0.00249, so we can write,

$\begin{matrix}\begin{matrix}{{\ln\left( \frac{2l_{MIN}}{w + t} \right)} = {{{{0.0}0249} - {0.5}} = {{- {0.4}}9751}}} \\\begin{matrix}{{e^{-}}^{0.49751} = {{0.6}08}} & {e^{- 0.5} = {{0.6}0653}}\end{matrix}\end{matrix} & (1.9)\end{matrix}$

A solution of (1.9) for l_(MIN) will give the minimum value for l forwhich (1.4) will be valid. Below this l_(MIN) value Equation (1.4) willgive a negative inductance value, which is not physical. Therefore (1.4)is valid only for l>l_(MIN). The value of l_(MIN) depends on the sum ofwidth w and the thickness t and can be given as,

$\begin{matrix}{l_{MIN} = {{{\frac{w + t}{2}e^{{f_{1}({w,t})} - 0.5}} \cong {\frac{w + t}{2}e^{- 0.5}}} = {0.30326 \cdot {\left( {w + t} \right).}}}} & (1.1)\end{matrix}$

Similar calculations can be done to obtain l_(MIN) for othercross-section geometries.

The Neumann integral formulation of mutual inductance for straightinfinitely thin filaments (the “Filament Method”) as shown in FIG. 1.3is,

$\begin{matrix}{M = {\int\limits_{P_{u}1}^{P_{u}2}{\int\limits_{P_{v}1}^{P_{v}2}{\frac{\overset{\rightarrow}{u} \cdot \overset{\rightarrow}{v}}{r}{ds}_{u}{ds}_{v}}}}} & (1.11)\end{matrix}$

where, u and v are unit vectors on each filament.

Performing the integral (1.11) for two parallel lines with a length of Zand spacing d as shown in FIG. 1.3 .1 leads to,

$\begin{matrix}{M = {0{\text{.002} \cdot {10^{-}}^{6} \cdot l \cdot {\left\lbrack {{\ln\left( {\frac{l}{d} + \sqrt{1 + \frac{l^{2}}{d^{2}}}} \right)} - \sqrt{1 + \frac{d^{2}}{l^{2}}} + \frac{d}{l}} \right\rbrack.}}}} & (1.12)\end{matrix}$ l, d → cm → M[H]

To calculate the mutual inductance between any parallel two arbitrarycross-sectional geometries with the same length l—under the assumptionof uniform current density distribution—one calculates the (g.m.d)between the two cross-sectional geometries and substitutes the g.m.d.value as the variable d in the filament formula in (1.12).

The “Filament Method” can be generalized for any filament arrangementand placement in three-dimensional space by double integrals in theNeumann integral formulation given by (1.11), again under the assumptionof constant current density distribution. For n coupled inductors, wheren>1, given as a multi-inductor system shown in FIG. 1.3 .2, one candefine an (nxn) “Inductance Matrix” L as,

$\begin{matrix}{{L\left( {n \times n} \right)} = {{\begin{bmatrix}{L_{1,1}L_{1,2}L_{1,3}\ldots} & {L_{.{,.}}L_{1,{n - 1}}L_{1,n}} \\{L_{2,1}L_{2,2}L_{2,3}\ldots} & {L_{.{,.}}L_{2,{n - 1}}L_{2,n}} \\{L_{3,1}L_{3,2}L_{3,3}\ldots} & {L_{.{,.}}L_{{3.n} - 1}L_{3,n}} \\\ldots & \\\ldots & \\{L_{{n - 1},1}L_{{n - 1},2}L_{{n - 1},3}\ldots} & {L_{.{,.}}L_{{n - 1},{n - 1}}L_{{n - 1},n}} \\{L_{n,1}L_{n,2}L_{n,3}\ldots} & {L_{.{,.}}L_{n,{n - 1}}L_{n,n}}\end{bmatrix}L_{i,j}} = {{L_{j,i}{and}L_{i,i}} > 0}}} & (1.13)\end{matrix}$

The diagonal matrix entries of (1.13) are self-inductance values, whichare always positive. The off-diagonal entries can be positive, negativeor zero. The Inductance Matrix L is ALWAYS symmetric as shown.

A quantity that is very important in a mutual inductance matrix forsatisfying the passivity requirement of the system is the coupling ratioK, which is expressed as,

$\begin{matrix}{K_{i,j} = {{\frac{L_{i,j}}{\sqrt{L_{i,i} \cdot L_{j,j}}}{and}{❘K_{i,j}❘}} < 1}} & (1.14)\end{matrix}$

where K_(ij), L_(ij), L_(i,i) and L_(jj) are the coupling ratio, selfinductances of elements i and j, the mutual inductances betweeninductances i and j respectively.

For an inductance matrix L resulting from same cross-sectional geometryand lengths will give equal self inductances at the diagonals. Havingthe same values in the diagonal of the inductance matrix (1.14) statesthat,

|K _(i,j)|<1→L _(i,j) ≤L _(i,i)  (1.15)

In other words, for this case the mutual inductances in the inductancematrix can never exceed the self-inductance values! This is a veryimportant concept that allows for the derivation and realization of theinvention herein disclosed.Inductance for a Rectangular Cross-Section Having a Constant area S andUnder Constant Current Density J_(MAX) Across its Cross-Section

As will be explained hereinafter, embodiments of the invention employ arectangular cross-section. One must always comply with minimumcross-section requirements when designing an inductor for anyapplication. Sources for this minimum cross-sectional area can be manyand are listed below, depending on the application;

i) For on-chip power converter applications [3-9], a large DC currentmust flow through the inductor, at least in the order of several Amperes(1-50 A) [3-9]. This imposes a minimum cross-sectional area S_(MIN)requirement on the inductor design due to the electro-migration currentdensity limit that cannot be exceeded. This electro-migration currentdensity limit JEM is a function of the chip metallization process. As anexample, for a typical Aluminum/Si alloy metallization, JEM is in theorder of 10⁵ to 2.10⁶ A/cm². For a given maximum DC currentspecification, one cannot exceed the current density imposed by theelectro-migration current density limit, therefore one cannot make theinductor cross-sectional area smaller than S_(MIN). S_(MIN) can becalculated as,

$\begin{matrix}{S_{MIN} = \frac{I_{{DC}{MAX}}}{J_{EM}}} & (1.16)\end{matrix}$

where I_(DCMAX) is the maximum DC current determined by the designspecification.

Table [1] shows the typical cross-sectional areas that need to besatisfied for some practical I_(DCMAX) values common in today's powermanagement and FIVR work.

TABLE 1 Inductor Current Determined Cross-Sectional Area, t_(OPT0) andt_(OPT) for 100 MHz and 200 MHz for Inductor Values of 20, 40, 60 and 80nH. I, Current (A) Cross-Section t_(OPT) (μ) t_(OPT) (μ) t_(OPT) (μ)t_(OPT) (μ) J_(max) = 2.10⁶ Area L = 20 nH L = 40 nH L = 60 nH L = 80 nHA/cm² S (μ²), t_(OPT0) (μ) f = 100 MHz f = 100 MHz f = 100 MHz f = 100MHz 1 50, 7.07 27.63 28.83 30.08 30.08 5 250, 15.81 107.9 117.5 122.6122.6 10 500, 22.36 195.9 213.3 222.6 222.6 20 1,000, 31.62  355.6 387.3404.1 421.7

ii) When designing an inductor, one always has a resistance RIND inseries with the inductor. Again, in power converter applications, theefficiency of a step-down or buck converter is closely related to the DCresistance of the inductor R_(IND) and cannot exceed an R_(IND_MAx)value that is a small load-dependent quantity which can be given as,

$\begin{matrix}{R_{IND\_ MAX} = \frac{V \cdot \left( {1 - \eta} \right)}{I \cdot \eta}} & (1.17)\end{matrix}$

where V, η and I are the DC voltage output, power conversion efficiencyand load current, respectively. As an example, a 1V DC supply with a 1 Aload is equivalent to 1Ω load resistance. For a 90% efficiency goal in abuck converter design with no switching losses, one has to keep the DCresistance R_(IND_MAX) of the inductor less than 0.11Ω. A typical“conservative” FIVR on-chip buck converter specification is 10 A at 1V.As can be seen for this case the DC resistance of the inductor RIND MAXhas to be less than 0.011Ω regardless of the inductor value. This can bea major challenge. FIG. 2.33 shows a graph of (1.17) illustrating theallowable maximum DC resistance R_(IND_MAX) as a function of output DCcurrent for 1 VDC output for 95, 90, 85 and 80% theoretical maximumefficiencies in any type of DC/DC converter topology regardless of theinductor values.

iii) The circuit topology of LNA (Low Noise Amplifier) IC designs canemploy several on-chip inductors, transformers or Baluns(Balanced-un-Balanced). Since the thermal noise voltage generated in aresistor is proportional to the square root of its resistance at theoperating frequency, the resistance value is an important contributor tothe overall noise figure of the LNA [31]. Therefore one needs to keepthe resistance of these passive elements less than an R_(MAX) value atthe operating frequency. The R_(MAX) value at a given frequency for thiscase is not straight-forward to calculate, but it is related to its DCresistance, and the “smaller the better” principal always applies. TheAC-resistance-to-DC-resistance relation is given for rectangularcross-sections hereinafter.

iv) For VCO/PLL (Voltage Controlled Oscillators/Phase Lock Loop)applications the phase noise is inversely proportional to the square ofthe Q value of the inductor. Q of an inductor. It is a complex quantityto calculate as a function of frequency, but it is very closely relatedto the AC resistance of the inductor. AC resistance is related to the DCresistance R_(DC), so again the “smaller the better” rule applies forR_(MAX)!

v) The power consumption limit on the inductor is another design factorin determining the R_(MAX) value at the frequency of operation. It isgiven by relation,

$\begin{matrix}{R_{MAX} = \frac{P_{MAX}}{I_{MAX}^{2}}} & (1.18)\end{matrix}$

where P_(MAX) is the maximum power consumption specification at a givenoperating frequency on the inductor, where R_(MAX) is the frequencydependent resistance of the inductor. Again R_(MAX) is related toR_(DC).

As can be seen, there is one factor that directly defines the S_(MIN) ofthe inductor design and there are an additional four factors that areindirectly related to it through the DC resistance formula of a prism,given as,

$\begin{matrix}{R_{DC} = {{\rho\frac{l}{S}} = {\rho\frac{l}{w \cdot t}}}} & (1.19)\end{matrix}$

where l, ρ, S, w and t are the length, resistivity, cross-sectionalarea, width and thickness for a rectangular cross-section straightconductor.

In any kind of inductor design one starts with a straight wire andgenerates geometries that give mutual inductances such that theirpresence increases the inductance compared to its straight conductorinductance value. Therefore, there is a need to formulate a geometricalconfiguration that provides for high mutual couplings with the propersign in the designed structure, such as in coils and spiral inductors.Calculating even R_(DC) for any kind of inductor design such as spiralor coil inductors is not straightforward, but it is straightforward tocalculate it for a straight conductor. That is a good point to start theanalysis.

Optimum Width/Thickness Relation for a Desired Inductor Value L for aGiven Cross-Sectional Area S for a Rectangular Cross-Section StraightConductor Giving Highest Possible Q for Uniform Current DistributionAssumption Across its Cross-Section

The “exact” result of the starting analysis under this model isincorrect, but it is a suitable starting point for the optimization ofan inductor, and it conveys the key points and the thought process ofthe invention very quickly and clearly, as hereinafter shown.

The problem at this stage is to find out if one can design an inductorwith a desired value L with minimum resistance R for a givencross-sectional area S. If there is such a width/thickness combination,using this combination will give the highest Q possible for a straightinductor. Ignoring capacitive and high frequency effects on the qualityfactor Q, the simple case is given as,

$\begin{matrix}{Q_{LRDC} = \frac{L\omega}{R}} & (1.2)\end{matrix}$

As can be seen, the Q formulation (1.20) assumes that both theresistance and the inductance of the straight wire is frequencyindependent. Therefore the subscript DC in (1.20) indicates the lowfrequency Q, where L and R are not taken as functions of frequency andwhere there are no capacitance effects. These effects will be brought into the analysis later.

First, inductor relation for a rectangular cross-section, as given in(1.4) as a function of given area S is,

$\begin{matrix}{L_{RECT} = {0{\text{.002} \cdot 10^{- 6}}{l \cdot {\left\lbrack {{\ln\left( \frac{2l}{\frac{S}{t} + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack.}}}} & (1.21)\end{matrix}$

As can be seen in (1.21) the width w, in (1.4) is just replaced by Sit.Close examination of (1.21) suggest that one can expect a thickness t,which gives the shortest length l for a desired inductance value L forany given cross-sectional area S. Ideally one needs to differentiate(1.21) with respect to t, equate it to zero and solve t to perform thistask. This requires several intermediate differentiation steps in theprocess. It can be done with simpler and shorter intermediate math andwill be more conclusive, as well as get the point across more quicklyand clearly. (We will have to take the longer approach for Qoptimization later, so no need to complicate the matter at this veryfirst stage!)

Let the u(t,S) function, which is the denominator term in the logfunction given in (1.21) be defined as,

$\begin{matrix}{{{u\left( {t,S} \right)} = {{w + t} = {\frac{S}{t} + t}}}.} & (1.22)\end{matrix}$

The t value, which gives the minimum for u(t,S), will satisfy this“minimum length for a given inductance value of L for any given S”condition by differentiating it with respect to t as,

$\begin{matrix}{\frac{du}{dt} = {1 - \frac{S}{t^{2}}}} & (1.23)\end{matrix}$

and equating (1.23) to zero and solving gives,

$\begin{matrix}{\frac{du}{dt} = {\left. 0\rightarrow t \right. = {\sqrt{S}.}}} & (1.24)\end{matrix}$

As can be seen in (1.24), there is an optimal thickness t, for a given Sand moreover the width w and thickness t are equal, which is directlythe result of area relation, giving,

w=t=√{square root over (S)}.  (1.25)

The advances in the art that form the basis of the present invention cannow be identified. One conclusion is this: To minimize the length of aconductor for a desired inductor value of L for a given cross-sectionalarea S in a rectangular cross-sectional geometry, one would need asquare cross-section with sides given as (1.25)!

FIG. 1.4 shows the u(t,S) for several cross-sectional areas where S=50,250, 500 and 1000μ², which are calculated from I_(MAX)=1, 5, 10 and 20.A design specification is contemplated using J_(EM)=2.10⁶ A/cm² to havesome “real” cases related to FIVR (Fully Integrated Voltage Regulator)cases in the analysis. As can be seen in FIG. 1.4 the u(t,S) functionhas an asymptote at t=0 and starts from infinity, goes through a minimumas calculated in (1.25) and approaches the u=t function, againasymptotically, with increasing t from zero to infinity. With theuniform current density distribution approximation, designing theinductor with a square cross-section and sides given as (1.25) will givethe highest possible Q because it will have the shortest length givingminimum resistance R given by (1.18) for the desired L and given S! This“minimum length” result is shown in FIG. 1.5 .1 which are plots ofstraight conductor inductor length versus thickness having constantcross-sectional areas S=50, 250, 500 and 1000μ². The target inductancevalues in the FIG. 1.5 .1 are 20, 40, 60 and 80 nH, which are considered“very large” value inductors for any on-chip inductor designs. For highfrequency LNA/VCO designs, with operating frequencies greater than 900MHz, the typical targeted on-chip inductor values are between 1 to 5 nH.On the other hand, in studies of the current hot topic of integratingthe buck converter on a processor, or FIVR (Fully Integrated VoltageRegulator), the investigated low DC resistance and high Q inductorvalues are in 10-60 nH range, operating at 20-200 MHz [3-9]. As can beseen, in the simplest Q relation given in (1.20), designing high Qinductors for operating frequencies in the order of 20-200 MHz, alongwith DC resistances in the order of 0.020 ohms, such as in FIVR work,makes the challenge much greater than for higher frequency RFIC inductordesigns.

The typical metal thickness in IC (Integrated Circuit) technology isless than a micron. Very few IC processes provide higher than about 3μto 4μ metal thickness and only at the top metal layer (M5) with 2.8μwidth and spacing. As can be seen, the thicknesses obtainable from thisvery first simplified analysis are much larger than any metalthicknesses provided in any known IC process technology, which are shownin Table [1] as t_(OPT0).

Combining the resistance formulation given by (1.19) for uniform currentdensity distribution, one can plot the Q versus thickness for given Land S values as well and see the peak Q being at the calculatedthicknesses. As noted above, the simplified analysis shown here is veryimportant in showing that there is an optimal w and t as a function of agiven cross-sectional area S for maximizing Q and it is also independentfrom the targeted inductance value. Closer examination howeverdemonstrates that this “qualitative” result is misleading. It ismisleading in that the resulting cross-sectional geometry, which is saidto be a square and is dependent on the given area S, is given as in(1.25). The misleading result comes from the “constant current density”assumption given by (1.19) for the resistance which cannot be satisfiedat operating frequencies of the inductors under consideration due toskin and proximity effects. The following section includes anexplanation of the non-uniform current density distribution effects in arectangular cross-section with some detail, taking the skin effect intoconsideration.

Optimum Width/Thickness Relation for a Desired Inductor Value L for aGiven Cross-Sectional Area S for a Rectangular Cross-Section StraightConductor Giving Highest Possible Q for Non-Uniform Current DistributionAssumption Across its Cross-Section

The general analysis of this problem can be tackled by solving Maxwell'sequations for a sinusoidal wave assumption which can be reduced to thesolution of the Helmholtz's equation in non-uniform media. Just for theelectric field E, Helmholtz wave equation in complex form can be writtenas [24, 25],

$\begin{matrix}{{{\nabla^{2}E} + {\omega^{2}\mu{\varepsilon\left( {1 - {j\frac{\sigma}{\omega\varepsilon}}} \right)}E}} = {{0{where}j} = {{\sqrt{- 1}{and}\omega} = {2\pi f}}}} & (1.26)\end{matrix}$

where μ, ε, σ, ω and ƒ are the magnetic permeability, dielectricconstant, conductivity, angular frequency and frequency, respectively. Asimilar equation can be written for the magnetic fields H and it can beshown that they are related [24,25]. A complete spiral inductor analysiscan be done by solving (1.26) for the three dimensional spiral geometry(which is possible only numerically). Employing Maxwell's equation whichleads to Helmholtz wave equation (1.26) in three dimensions, one cancalculate electric and magnetic fields and the non-uniform currentdensity distribution in any point of any of the winding regions [10-23].This is a complex analysis, but looking at the results of the complexsimulations and using some common electromagnetic sense, the analysiscan be simplified. Here we will first focus on the principles and theimportant analytical results of the solution of the simplified Helmholtzwave equation for plane waves, which is applicable to the spiralinductor regions in the interior of its winding as shown in a verysimple spiral inductor geometry shown in FIG. 1.6 .1. There are severalchoices of forcing the current I in and getting it out of the spiralinductor. Several of many choices are shown with pad locations atIN_(CENTER), OUT_(CENTER), IN_(SIDE) and OUT_(SIDE) locations in FIG.1.6 .1.

The cross-sectional geometry of the spiral inductor structure shown inFIG. 1.6 .1 is given in FIG. 1.7 .1. The cross-section geometry isobtained by having cut planes passing from x=x_(CUT) or y=y_(CUT) linesperpendicular to the (x,y) plane as shown in FIG. 1.6 .1. For simplicitythe shown spiral structure has a square inner space with a dimension ofd_(IN) and has its windings have a constant width w and spacing s. Sincethis invention defines a spiral inductor with a large aspect ratio metaland metal spacing, the FIG. 1.7 .1 points to this feature of theinvention as is shown very clearly. Known spiral inductor structureshave metal winding widths much larger than their thicknesses as shown inFIG. 1.9 .1 due to the metallization achievable in any IC, PCB orceramic processes. If one can build an interposer structure withmetallization rules based on what is required for these inductors andplace it on top of the IC or under it with the shortest connectionspossible to the IC, this will give a very practical solution to theproblem.

FIG. 1.7 .1 also shows another key aspect of the invention in theinterposer structure having very thick metal along with very high aspectratios for metal lines and spacing, which has its connections to theoutside circuit from both sides of the structure. FIG. 1.7 .1 shows thisvery preferable capability for power management IC applications. Inthese circuits, as in Buck converters, the inductor is placed betweenthe supply and the switching network [3-9,30]. The switching network isbuilt into the IC and the inductor is placed between the switch and thesupply pin [3-9, 30]. According to this invention, however, the inductoris built on the interposer structure and is connected to the IC from thebottom of the interposer as shown in FIG. 1.7 .2 with a ball, a standardtechnique used in ball grid arrays, and it will connect to the supplypin from the top. As can be seen having the interposer right on top ofthe IC with connections available on both sides of the interposer savesvery valuable space. On the other hand if the interposer access wereonly possible from one side, a very valuable area will be lost inpackaging. In fact for this case, placing the interposer right on top ofthe chip will not even give any area saving advantage for any powermanagement IC applications.

In some other applications, such as in RFIC, PLL and VCO's, both of thepins of the inductor have to be connected to the IC. This is illustratedin FIG. 1.8 .1 and FIG. 1.8 .2. In this case the high value, high Qinductor is built on the interposer and, through the two balls on thebottom of the interposer, it is connected to the desired pads in the IC.Since there can be multiple inductors built on the interposer for bothtypes of circuits, both sides of the interposer can be used to connectthe IC through the interposer to the outside world with additionalpassive circuits on the interposer with improvements in performance andpacking density without having any area penalty.

FIG. 1.9 .1 and FIG. 1.9 .2 show, according to the invention, two kindsof rectangular cross-section spiral winding arrangements, both havingthe same cross-sectional areas S, but placed differently. The toparrangement shown in FIG. 1.9 .1, which is noted as subscript 1,represents the typical arrangement in prior art spiral inductor designshaving width w larger than thickness t or w>>t. This choice is dictatedby the need for a small R_(DC) for a given thin metal thickness t, so itis something one cannot avoid for a practical design specification. Thebottom arrangement, shown in FIG. 1.9 .2, noted with the subscript 2,and made according to this invention, shows conductors having thicknesst larger than the width w, written as t>>w. To be able to realize thisstructure, one must employ a process with metal thicknesses fairly largecompared to what is available today as shown in Table [1]. As can beseen in this arrangement the conductors are intentionally placed withlonger dimensions facing each other and shorter dimension facing towardthe ground plane or the substrate. Width w and the thickness t aredimensions defined along x and z directions of the coordinate system.The electromagnetic field propagation in the region is assumed to bealong y direction, coming out of the plane of FIG. 1.9 .1 and FIG. 1.9.2 and with no electromagnetic field components along the propagationdirection, which is known as TEM mode. As a result the electric andmagnetic waves only have x and z components as shown in FIG. 1.9 .1 andFIG. 1.9 .2 which converts the problem into a two-dimensional problem.

When an electromagnetic wave enters a conductive region, such as intothe windings of the spiral cross-section shown in FIG. 1.9 .1 and FIG.1.9 .2 from the lossless dielectric region between the windings andbetween the conductors and the ground plane, it will attenuateexponentially with an attenuation constant α along its propagationdirection into the conductor as a result of the solution of theHelmholtz wave equation (1.26). Solving (1.26) for plane waves, thisattenuation constant α for electric and magnetic fields into theconductors will be [24, 25],

$\begin{matrix}{\alpha = {\omega\sqrt{\frac{\varepsilon\mu}{2}\left( {\sqrt{1 + \left( \frac{\sigma}{\omega\varepsilon} \right)^{2}} - 1} \right)}}} & (1.27)\end{matrix}$

under a “good conductor” approximation which is defined as,

$\begin{matrix}{\frac{\sigma}{\omega\varepsilon}\operatorname{>>}1.} & (1.28)\end{matrix}$

Equation (1.27) simplifies to the well-known “skin depth” [24, 25] givenas,

$\begin{matrix}{\delta = {\frac{1}{\alpha} = {\sqrt{\frac{2}{\omega\sigma\mu}} = {\sqrt{\frac{\rho}{\pi\sigma\mu f}}.}}}} & (1.29)\end{matrix}$

FIG. 1.10 show the skin depth as a function of frequency between 10Hz-10 GHz for Cu, Al and Si having 1 and 10 ohm-cm resistivity. Sincethe trend and the desire is to integrate inductors on a silicon basematerial as the substrate, FIG. 1.10 includes the skin depth in typicalSi substrate resistivity ranges of silicon. In all cases, relativemagnetic permeability is taken as μ_(r)=1 giving μ=1.25663.10⁻⁸ H/cm in(1.29). In (1.29) the resistivity of Cu and Al is taken as 1.75.10⁻⁶ and2.73.10⁻⁸ ohm-cm respectively, and the skin depth as a function offrequency is plotted and shown in FIG. 1.10 . As can be seen, the lowerthree curves (Cu, Al, 1 ohm-cm Si) skin depth varies inverselyproportional with the square root of the frequency in the log scalelinearly at the entire frequency range, which tells us that the “goodconductor” approximation (1.28) is valid for these cases. On the otherhand, at frequencies higher than 2 GHz the 10 ohm-cm resistivity Simaterial shows some small amounts of flattening. This can be seen atmuch lower frequencies for higher resistivity Si. In other words the“good conductor” approximation holds very well for Cu, Al and for 1ohm-cm resistivity Si in the entire frequency range, but not as good for10 ohm-cm and higher resistivity Si above 2 GHz. Therefore it is alwaysa good practice to check if the “good conductor” approximation holds inthe frequency range of interest.

From the calculated electric and magnetic field distributions in theconductor regions of FIG. 1.9 .1 and FIG. 1.9 .2 one approximates thenon-uniform current density distributions in the spiral windings, whichwill result in they direction current flow as observed in spiralinductor structures.

One can speculate on three possible forms of solutions for thenon-uniform current density distribution in the spiral windings:

i) Current Density Distribution is Uniform at the Surface of theConductors.

This solution cannot apply to the majority of the known spirals whichhave w>>t. If the width and thicknesses (w,t) are not in the same orderand if the distance d_(G) between the ground plane and winding is not onthe same order of magnitude as the spacing s, the external fielddistribution on the conductor surface will be far from uniform andtherefore cannot support this boundary condition. Therefore it only hasits merit for square cross-sections in conductors having similar s andd_(G). This mode has an analytical solution for circular cross-sections[24] using Bessel functions (Ber, Bei), but for a rectangularcross-section, one needs a numerical solution of the Helmholtz waveequation where an analytical solution does not exist.

On the other hand, if t>>w and s<<d, and there are vertically stackedinductor winding structures which have w>>t, d (as shown forillustrative purposes in FIG. 1.12 ), in a structure formed according tothe invention, then numerical simulations suggest that this boundarycondition—namely, uniform surface current density distribution—becomes afairly good approximation of the actual current density distribution.FIG. 1.12 only shows one side of the vertically stacked inductor windingstructures where the layers are connected through vias upward anddownward, which is basically a structure of coils made in a planarprocess with a large number of metal layers.

ii) Majority of the Fields are Confined Between the Bottom of theConductors and the Ground Plane.

This solution applies fairly well for PCB (Printed Circuit Boards),ribbons and the majority of the on-chip spiral inductors where w>>t ands>d_(G). For this case a fairly good analytical derivation of thecurrent density distribution can be given, which is once again in linewith the solution of the Helmholtz wave equation for the exactstructures. This type of current density distribution in the windings isreferred as the “single sided solution” in this disclosure. Thereforethe straightforward analytical analysis will be given once and for thecase related to the invention below.

iii) Majority of the External Fields are Between the Windings.

The solution in this case is applicable to one of the embodiments of theinvention explained in this work given as, t>>w, d_(G)<<s. Thisapproximation also applies to the structure where longer sides of thewindings are along the x axis, but are stacked as shown in FIG. 1.12 , acomplete 90° flip of the structure of a preferred embodiment of theinvention.

Single Sided Current Density Distribution Assumption and its Impact onR_(AC)/R_(DC)

Consider an analytical derivation for the conductor cross-section shownon the right of FIG. 1.9 .2, which is referred as “single sidedsolution”. For now assume the metal width of the windings shown in FIG.1.9 .2 is infinite, in other words w₂→∞ and the current density at x=0is J₀. For the given current density J₀ at the conductor surface (x=0)the solution will give [26-29],

$\begin{matrix}{{J_{AC}(x)} = {J_{0}e^{- \frac{x}{\delta}}}} & (1.3)\end{matrix}$ ${{where}\delta} = {\sqrt{\frac{2}{\sigma\omega\mu}}.}$

In reality the electric and magnetic fields can enter from both sides ofthe spiral windings. For this case the problem becomes more complex forspiral inductor current density calculations in its windings and it isreferred as the “complete solution” and will be discussed herein below.

Integrating the current density relation (1.30) from 0 to the width wgives the total current flowing in the conductor and is formulated as,

$\begin{matrix}{I_{AC} = {{\int\limits_{0}^{w}{J_{0}e^{- \frac{x}{\delta}}dx}} = {J_{0}{{\delta\left( {1 - e^{- \frac{w}{\delta}}} \right)}.}}}} & (1.31)\end{matrix}$

On the other hand the DC current, where the current density is uniform,is given as,

I _(DC) =J ₀ w.  (1.32)

If the windings were infinitely wide, then the total AC current as aresult of (1.31) becomes,

1im _(w→∞)(I _(AC))=J ₀δ.  (1.33)

As can be seen in (1.33), the AC current does not go to infinity as itwould in the DC current case! We can ratio the AC current expression(1.31) to the DC current relation (1.32) and get the frequency-dependentAC resistance as,

$\begin{matrix}{\frac{R_{AC}}{R_{DC}} = {\frac{I_{DC}}{I_{AC}} = {\frac{w}{\delta \cdot \left( {1 - e^{- \frac{w}{\delta}}} \right)}.}}} & (1.34)\end{matrix}$

Rearranging variables in (1.34) gives,

$\begin{matrix}{\frac{R_{AC}}{R_{DC}} = {{\frac{u}{\left( {1 - e^{- u}} \right)}{where}u} = {\frac{w}{\delta}.}}} & (1.35)\end{matrix}$

The expression (1.35) is the Bernoulli Generation Function [27] for n=1defined as,

$\begin{matrix}{n = {\left. 1\rightarrow\frac{t}{\left( {1 - e^{- t}} \right)} \right. = {\sum\limits_{m = 0}^{\infty}{B_{m}{\frac{\left( {- t} \right)^{m}}{m!}.}}}}} & (1.36)\end{matrix}$

In shorthand, we refer to (1.36) as B(u). B(u) is a Bernoulli functionthat will give only positive values for −∞<u<∞ and 1 at u=0 as shown inFIG. 1.11 .1. The region of interest according to this invention is theu>0 region and as can be seen, (1.35) becomes linearly dependent on thew/δ ratio after w/δ>5, giving,

$\begin{matrix}{u = \left. {\frac{w}{\delta} > 5}\rightarrow{R_{AC} \approx {R_{DC}{\frac{w}{\delta}.}}} \right.} & (1.37)\end{matrix}$

As an example for u=w/δ=10, one can right away say R_(AC)/R_(DC) will bevery close to 10 without any calculations. If the w/δ ratio is small, bysolving the nonlinear equation (1.35) for a desired acceptableR_(AC)/R_(DC) value at a given frequency one can determine the width was a function of δ. FIG. 1.11 .2 shows this process graphically veryclearly. FIG. 1.11 .2 shows that to have R_(AC)/R_(DC)=2 the width w hasto be 1.6×δ. If the width is set equal to δ, the R_(AC)/R_(DC) will be1.6 at that frequency, which is very suitable for an inductor design.

The top curve in FIG. 1.21 .1 shows R_(AC)/R_(DC) as a function of u,which is the w/δ ratio. FIG. 1.21 .2 shows R_(AC)/R_(DC) as a functionof width at 100 MHz the for Cu. As can be seen from askin-depth-versus-frequency curve shown in FIG. 1.10 and its relation toR_(AC)/R_(DC) as shown in FIG. 1.21 .1 and FIG. 1.21 .2, one has to keepthe width of the windings not much larger than the skin depth. Thesecorrespond to fairly small widths for the vertical metal arrangementshown in FIG. 1.9 .2. The same is true for the thickness for ahorizontal metal arrangement as shown in FIG. 1.9 .1. In this casehaving a thick metal compared to skin depth at that frequency will notimprove the R_(AC)/R_(DC) significantly but will only help to reduce theDC resistance. Since we need to maintain a given cross-sectional area S,the only solution is to increase the thickness t of the vertical metalarrangement as shown in FIG. 1.9 .2.

Table [2.0] shows skin depth as a function of typical frequencies forcopper and aluminum. As will be noted the skin depth is on the sameorder of magnitude as the width dimension of the inductor.

TABLE 2 Skin Depth (δ) as a Function of Selected Typical Frequencies forCopper (1.75 · 10⁻⁶ Ω · cm) and Aluminum (2.73 · 10⁻⁶ Ω · cm) andCritical Width or Thicknesses to Maintain R_(AC)(f)/R_(DC) = 2. δ(Al) ×1.6 δ(Al) × 3.85 δ(Cu) × 1.6 δ(Cu) × 3.85 [μ] [μ] [μ] [μ] R_(AC)/R_(DC)= 2 R_(AC)/R_(DC) = 2 R_(AC)/R_(DC) = 2 R_(AC)/R_(DC) = 2 f δ(Al)“Single Sided” “Complete” δ(Cu) “Single Sided” “Complete” [MHz] [μ]Solution Solution [μ] Solution Solution 25 16.63 26.608 64.025 13.2221.152 50.897 50 11.76 18.816 45.276 9.348 14.957 35.989 100 8.31613.306 32.017 6.610 10.576 25.449 200 5.880 9.408 22.638 4.674 7.47917.995 900 2.772 4.435 10.672 2.203 3.525 8.483 1,200 2.401 3.841 9.2421.908 3.053 7.347 1,575 2.095 3.353 8.067 1.666 2.665 6.414 2,400 1.6972.716 6.533 1.349 2.159 5.195 5,200 1.153 1.845 4.439 0.916 1.467 3.529

Single Sided Current Density Distribution Assumption and its Impact on Q

In designing a high Q inductor, it is important to calculate the R_(AC)besides the R_(AC)/R_(DC). The AC resistance can be calculated as,

$\begin{matrix}{R_{AC} = {{R_{DC}\frac{u}{\left( {1 - e^{- u}} \right)}{where}u} = {\frac{w}{\delta}.}}} & (1.38)\end{matrix}$

For a given length 1, substituting the DC resistance relation (1.18) in(1.38) the AC resistance becomes,

$\begin{matrix}{R_{AC} = {{\rho\frac{l}{w \cdot t}\frac{u}{\left( {1 - e^{- u}} \right)}{where}u} = {\frac{w}{\delta} = {w{\sqrt{\frac{\mu\omega}{2\rho}}.}}}}} & (1.39)\end{matrix}$

With some arithmetic manipulation, (1.39) gives,

$\begin{matrix}{R_{AC} = {{\frac{l}{t}\sqrt{\frac{\rho\mu\omega}{2}}\frac{1}{\left( {1 - e^{- u}} \right)}{where}u} = {\frac{w}{\delta} = {w{\sqrt{\frac{\mu\omega}{2\rho}}.}}}}} & (1.4)\end{matrix}$

As can be seen in (1.40) the width dependency AC resistance is very weakcompared to the DC resistance for large values of u!

Another interesting result is that the R_(AC) is no longer directlyproportional to resistivity ρ as in R_(DC). It becomes linearlyproportional to the square root of the resistivity and frequency.

Rearranging (1.40) gives,

$\begin{matrix}{\begin{matrix}{{R_{AC}\left( {\omega,l,t,w,\rho,\mu} \right)} = {\frac{l}{t}\sqrt{\frac{\rho\mu\omega}{2}}\frac{1}{\left( {1 - e^{- u}} \right)}}} \\{{{where}u} = {\frac{w}{\delta} = {w\sqrt{\frac{\mu\omega}{2\rho}}}}}\end{matrix}.} & (1.41)\end{matrix}$

As can be seen, making width w much wider than the skin depth δincreases the R_(AC)/R_(DC) ratio, but still helps to reduce the R_(AC).

Thus, making the width w wider does not give much reduction in the ACresistance. Rather it just wastes area and adds more capacitance to theinductor if not carefully calculated (as shown herein below)! It is abetter practice to increase the thickness rather than to increase thewidth for maintaining the same cross-sectional area S!

Ultimately we are interested in a width w of a conductor giving highestpossible Q at a given frequency while maintaining the samecross-sectional area S for a desired inductance value of L.

First one calculates the Q of a straight inductor, ignoring all thecapacitance effects, under the uniform current density distribution andnames it Q_(LRDC), which actually represents a very low frequency case.Applying Q relation given at (1.20) gives,

$\begin{matrix}{Q_{LRDC} = {\frac{0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack}}{\rho\frac{l}{w \cdot t}} \cdot {\omega.}}} & (1.42)\end{matrix}$

Doing some arithmetic manipulation gives,

$\begin{matrix}{Q_{LRDC} = {\frac{0{\text{.002} \cdot 10^{- 6} \cdot w \cdot t}}{\rho} \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack \cdot {\omega.}}} & (1.43)\end{matrix}$

As can be seen, the length l in the front of (1.42) cancels and (1.43)can be written in terms of S as,

$\begin{matrix}{Q_{LRDC} = {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack \cdot {\omega.}}} & (1.44)\end{matrix}$

Forcing the constant S condition to (1.44) as done earlier allows (1.44)to be written as,

$\begin{matrix}{Q_{LRDC} = {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \left\lbrack {{\ln\left( \frac{2l}{\frac{S}{t} + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack \cdot {\omega.}}} & (1.45)\end{matrix}$

Doing some arithmetic manipulation in the log expression as done earliergives,

$\begin{matrix}{Q_{LRDC} = {\frac{0{\text{.002} \cdot 10^{- 6}}S}{\rho} \cdot \left\lbrack {{\ln\left( \frac{2 \cdot l \cdot t}{S + t^{2}} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack \cdot {\omega.}}} & (1.46)\end{matrix}$

Ignoring the terms after the natural log expression in the bracketapproximates (1.46) as,

$\begin{matrix}{Q_{LRDC} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \left\lbrack {\ln\left( \frac{2 \cdot l \cdot t}{S + t^{2}} \right)} \right\rbrack \cdot {\omega.}}} & (1.47)\end{matrix}$

Finding the maximum or minimum of (1.47) with respect to t requiresdifferentiation of (1.47) as before. With the help of the followingvariable transformation,

$\begin{matrix}{u = {{\ln\left( \frac{2 \cdot l \cdot t}{S + t^{2}} \right)} = {\ln(v)}}} & (1.48)\end{matrix}$

and using the basic differentiation rule [26-29],

$\begin{matrix}{\frac{du}{dt} = {\frac{1}{v} \cdot \frac{dv}{dt}}} & (1.49)\end{matrix}$ where $v = \frac{2 \cdot l \cdot t}{S + t^{2}}$

gives,

$\begin{matrix}{\frac{\partial Q_{LRDC}}{\partial t} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \omega \cdot \left\lbrack {\left( \frac{S + t^{2}}{2 \cdot l \cdot t} \right) \cdot \frac{dv}{dt}} \right\rbrack}} & (1.5)\end{matrix}$

where,

$\begin{matrix}{\frac{dv}{dt} = {\frac{{2 \cdot l \cdot \left( {S + t^{2}} \right)} - {\left( {2 \cdot t \cdot l} \right) \cdot \left( {2 \cdot t} \right)}}{\left( {S + t^{2}} \right)^{2}}.}} & (1.51)\end{matrix}$

With further arithmetic manipulation,

$\begin{matrix}{\frac{dv}{dt} = {\frac{{2 \cdot l \cdot \left( {S + t^{2}} \right)} - {4 \cdot l \cdot t^{2}}}{\left( {S + t^{2}} \right)^{2}} = {\frac{{2 \cdot l \cdot S} - {2 \cdot l \cdot t^{2}}}{\left( {S + t^{2}} \right)^{2}}.}}} & (1.52)\end{matrix}$

Simplification of (1.52) gives,

$\begin{matrix}{\frac{dv}{dt} = {\frac{2 \cdot l \cdot \left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)^{2}}.}} & (1.53)\end{matrix}$

Substituting (1.53) in (1.50) gives,

$\begin{matrix}{\frac{\partial Q_{LRDC}}{\partial t} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \omega \cdot {\left\lbrack {\left( \frac{S + t^{2}}{2 \cdot l \cdot t} \right) \cdot \frac{2 \cdot l \cdot \left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)^{2}}} \right\rbrack.}}} & (1.54)\end{matrix}$

Straight forward arithmetic manipulation on (1.54) results in,

$\begin{matrix}{\frac{\partial Q_{LRDC}}{\partial t} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \omega \cdot \frac{1}{\left( {S + t^{2}} \right) \cdot t} \cdot {\left( {S - t^{2}} \right).}}} & (1.55)\end{matrix}$

Equating (1.55) to zero and solving it,

$\begin{matrix}{{\frac{\partial Q_{LRDC}}{\partial t} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \omega \cdot \frac{1}{\left( {S + t^{2}} \right) \cdot t} \cdot \left( {S - t^{2}} \right)}} = 0} & (1.56)\end{matrix}$

which gives the same result for t as done before with much shorterintermediate math giving,

t=√{square root over (S)} w=t=√{square root over (S)}.  (1.57)

Let us denote the optimal thickness given in (1.57) as t_(OPT0). Theadvantage of taking this approach is to be sure of the result, as wellas finding the peak Q at (1.57), giving,

$\begin{matrix}{{Q_{LRDC}({MAX})} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \left\lbrack {\ln\left( \frac{2 \cdot l \cdot \sqrt{S}}{S + S} \right)} \right\rbrack \cdot {\omega.}}} & (1.58)\end{matrix}$

Doing the arithmetic in the natural log expression gives this veryinteresting result,

$\begin{matrix}{{Q_{LRDC}({MAX})} \cong {\frac{0{\text{.002} \cdot 10^{- 6} \cdot S}}{\rho} \cdot \left\lbrack {\ln\left( \frac{l}{\sqrt{S}} \right)} \right\rbrack \cdot {\omega.}}} & (1.59)\end{matrix}$

Table [1] above shows t_(OPT0) for cross-sectional areas determined bydesired current values I without violating Al electro-migration currentdensity rules for various large value inductors at 100 MHz. As can beseen, even with the uniform current density assumption, which it will beshown to be incorrect, the required metal thickness values are verylarge compared to IC process metal thicknesses!

Applying the “Single Sided Current Density Distribution Assumption” inthe analysis done earlier by substituting the RAG term (1.41) in (1.42)gives,

$\begin{matrix}{{Q_{LRAC}.} = {\frac{0{\text{.002} \cdot 10^{- 6}}{l \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack}}{\frac{l}{t}\sqrt{\frac{\rho\mu\omega}{2}}\left( \frac{1}{1 - e^{- u}} \right)} \cdot {\omega.}}} & (1.6)\end{matrix}$

Similar arithmetic gives,

$\begin{matrix}{Q_{LRAC} = {0{\text{.002} \cdot 10^{- 6} \cdot t \cdot \left( {1 - e^{- u}} \right) \cdot \sqrt{\frac{2}{\rho\mu\omega}} \cdot \left\lbrack {{\ln\left( \frac{2l}{w + t} \right)} + {0.5} - {f_{1}\left( {w,t} \right)}} \right\rbrack \cdot {\omega.}}}} & (1.61)\end{matrix}$

Applying the same approximation to (1.61) gives,

$\begin{matrix}{Q_{LRAC} = {0{\text{.002} \cdot 10^{- 6} \cdot \sqrt{\frac{2}{\rho\mu\omega}} \cdot t \cdot \left( {1 - e^{- v}} \right) \cdot \left\lbrack {\ln\left( \frac{2l}{\frac{S}{t} + t} \right)} \right\rbrack \cdot \omega}}} & (1.62)\end{matrix}$

where v that appears on the exponent as a function of S becomes,

$\begin{matrix}{v = {\frac{w}{\delta} = {\frac{S}{t}{\sqrt{\frac{\mu\omega}{2\rho}}.}}}} & (1.63)\end{matrix}$

Differentiation of (1.62) requires longer work, but it will not yieldthe result as earlier. To apply the chain rule [26-29] fordifferentiation easily let the following functions be defined as,

$\begin{matrix}{{g_{1}(t)} = {{\left( {1 - e^{- v}} \right){and}{g_{2}(t)}} = {\ln{\left( \frac{2l}{\frac{S}{t} + t} \right).}}}} & (1.64)\end{matrix}$

Substituting the g₁(t) and g₂(t) in (1.62) gives,

Q _(LRAC) =a·t·g ₁(t)·g ₂(t)  (1.65)

where,

$\begin{matrix}{a = {{0.002 \cdot 10^{- 6} \cdot \omega \cdot \sqrt{\frac{2}{\rho\mu\omega}}} = {0{\text{.002} \cdot 10^{- 6}}{\sqrt{\frac{2\omega}{\rho\mu}}.}}}} & (1.66)\end{matrix}$

The derivatives of g₁(t) and g₂(t) functions with respect to t are,

$\begin{matrix}{\frac{\partial g_{1}}{\partial t} = {{e^{- v}\frac{\partial v}{\partial t}} = {e^{- v}\left( {{- \frac{S}{t^{2}}}\sqrt{\frac{\mu\omega}{2\rho}}} \right)}}} & (1.67)\end{matrix}$

and,

$\begin{matrix}{\frac{\partial g_{2}}{\partial t} = {\frac{1}{\left( {S + t^{2}} \right) \cdot t} \cdot {\left( {S - t^{2}} \right).}}} & (1.68)\end{matrix}$

The chain rule applied to (1.65) gives,

$\begin{matrix}{\frac{\partial Q_{LRAC}}{\partial t} = {a \cdot {\left\lbrack {{{g_{1}(t)} \cdot {g_{2}(t)}} + {t \cdot \left( {{\frac{\partial g_{1}}{\partial t}{g_{2}(t)}} + {{g_{1}(t)}\frac{\partial g_{2}}{\partial t}}} \right)}} \right\rbrack.}}} & (1.69)\end{matrix}$

To shorten Equation (1.69) to a manageable representation to perform theanalysis more clearly, (1.69) can be written as the sum of y(t) and z(t)functions given as,

$\begin{matrix}{\frac{\partial Q_{LRAC}}{\partial t} = {a \cdot \left\lbrack {{z(t)} + {y(t)}} \right\rbrack}} & (1.7)\end{matrix}$

where,

$\begin{matrix}{{y(t)} = {t \cdot {\left( {{\frac{\partial g_{1}}{\partial t}{g_{2}(t)}} + {{g_{1}(t)}\frac{\partial g_{2}}{\partial t}}} \right).}}} & (1.71)\end{matrix}$

The first function y(t) can be written explicitly as,

$\begin{matrix}{{y(t)} = {t \cdot {\left\{ {{{{e^{- v}\left( {{- \frac{S}{t^{2}}}\sqrt{\frac{\mu\omega}{2\rho}}} \right)} \cdot \ln}\left( \frac{2{lt}}{S + t^{2}} \right)} + {\frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right) \cdot t}\left( {\frac{S}{t}\sqrt{\frac{\mu\omega}{2\rho}}} \right)}} \right\}.}}} & (1.72)\end{matrix}$

With some arithmetic manipulation, (1.70) becomes,

$\begin{matrix}{{y(t)} = {\frac{S}{t}{\sqrt{\frac{\mu\omega}{2\rho}}\left\lbrack {{{{- e^{- v}} \cdot \ln}\left( \frac{2{lt}}{S + t^{2}} \right)} + \frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}} \right\rbrack}}} & (1.73)\end{matrix}$

and,

$\begin{matrix}{{z(t)} = {{\left( {1 - e^{- v}} \right) \cdot \ln}{\left( \frac{2{lt}}{S + t^{2}} \right).}}} & (1.74)\end{matrix}$

After some simple and straightforward arithmetic manipulations, (1.72)becomes simpler. Equating it to zero gives,

$\begin{matrix}{{{\left\lbrack {1 - {\left( {1 + \frac{b}{t}} \right) \cdot e^{- v}}} \right\rbrack\ln\left( \frac{2{lt}}{S + t^{2}} \right)} + {\frac{b}{t}\frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}}} = 0} & (1.75)\end{matrix}$

where,

$\begin{matrix}{b = {{S\sqrt{\frac{\mu\omega}{2\rho}}} = {v \cdot {t.}}}} & (1.76)\end{matrix}$

By using the relation (1.63) for v, (1.75) can be rewritten using “only”the variable v instead of variables of v and b as,

$\begin{matrix}{{{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack\ln\left( \frac{2{lt}}{S + t^{2}} \right)} + {v\frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}}} = 0.} & (1.77)\end{matrix}$

As can be seen, (1.77) is a nonlinear equation in t and can be solved“only” numerically. Before going into the solution of (1.77) it isuseful to see the functional behavior of v, which is an importantargument of (1.77) as a function of frequency ƒ. This analyticalinvestigation will also lead to very important results which are thebasis of claims of this application without even solving (1.77).

The first thing to note is that v will be “always” a positive number. Inaddition to that for any material when the signal in the medium is atzero frequency (ω=0), v will become zero (v=0). If we substitute somereal values into the v expression given in (1.63), such as for copper(Cu) resistivity, along with the numerical value of μ, v becomes,

$\begin{matrix}{v_{Cu} = {{\frac{S}{t}\sqrt{\frac{1.25 \cdot 10^{- 8} \cdot \omega}{2 \cdot 1.68 \cdot 10^{- 6}}}} = {{\frac{S}{t} \cdot 0.6099 \cdot 10^{- 1}}{\sqrt{\omega}.}}}} & (1.78)\end{matrix}$

It can be shown that (1.78) for Cu, frequencies above 42.7 Hz, thesquare root term exceeds 1 and increases with the square root of thefrequency. Now let's investigate the properties of the solution of(1.77) without solving it. It will show that the uniform currentdistribution solution is incorrect for frequencies ƒ>0!

i) The Optimal Thickness Tom′ Satisfying the Solution of (1.80) isGreater than S^(0.5), Giving High Aspect Ratio Conductor Cross-Section!

As can be seen, equation (1.77) is a summation of two terms. If wedivide both sides of (1.77) by v we get,

$\begin{matrix}{{{\frac{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack}{v}\ln\left( \frac{2lt}{S + t^{2}} \right)} + \frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}} = 0.} & (1.79)\end{matrix}$

As can be seen, the second term in (1.79) is,

$\begin{matrix}{\frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}.} & (1.8)\end{matrix}$

Equating (1.80) to zero and solving t will give the uniform currentexpression derived earlier, which will give the t_(OPT0)=S^(0.5)solution. It should be noted that (1.80) will be negative for t>S^(0.5)and positive for t<S^(0.5). Moreover (1.80) is zero for t=S^(0.5). Thefirst term in (1.79) is product of two functions,

$\begin{matrix}{{\frac{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack}{v}\ln\left( \frac{2lt}{S + t^{2}} \right)} > {0{for}{\ln\left( \frac{2lt}{S + t^{2}} \right)}} > {0.}} & (1.81)\end{matrix}$

Since the second logarithmic term in (1.81) is always a positive numberfor any practical length l, all we have to do is find the behavior ofthe first term of (1.81) with respect to v to determine the sign of(1.81). In other words, if the first term in (1.81) is proven to bealways positive for any v>0, the relation (1.81) will also become alwayspositive for any length l where we can apply the inductor formula (1.4).This can be done graphically very easily. FIG. 1.13 .1 shows the (1.82)u₁, as a function of v. As can be seen for all v>0,

$\begin{matrix}{{u_{1}\left( {\rho,\omega,t} \right)} = {\frac{\left\lbrack {1 - {\left( {1 + v} \right)e^{- v}}} \right\rbrack}{v} \geq 0.}} & (1.82)\end{matrix}$

For v=0 (ω=ƒ=0), (1.82) becomes a 0/0 type uncertainty, but it can beresolved by using L'Hospital's rule [26, 27]. Differentiating thedominator and denominator of it with respect to v and substituting v=0in the expression, gives 0 for v=0. It can also be shown that (1.82) hasa maximum for v>1 giving 0.2983 as its maximum value at v=1.8. Since itis proven that the first term of (1.79) is always positive for all v>0values, (1.81) is also positive for any v, l, S and t. In this case theequation (1.79) can only be satisfied if its second term (1.80) isnegative. This condition can mathematically given as,

$\begin{matrix}{{{\frac{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack}{v}\ln\left( \frac{2lt}{S + t^{2}} \right)} + \frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}} = 0} & (1.83)\end{matrix}$${{only}{if}\frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}} < 0$

If we rewrite (1.79) as,

$\begin{matrix}{{a + \frac{\left( {S - t^{2}} \right)}{\left( {S + t^{2}} \right)}} = 0} & (1.84)\end{matrix}$

where,

$\begin{matrix}{a = {{\frac{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack}{v}\ln\left( \frac{2lt}{S + t^{2}} \right)} \geq 0}} & (1.85)\end{matrix}$

and some arithmetic manipulation on (1.84) gives,

a(S+t ²)+(S−t ²)=0.  (1.86)

Finally, equation (1.84) becomes,

(a−1)t ² +S(1+a)=0.  (1.87)

Solving t from (1.87) gives,

$\begin{matrix}{{t^{2} = \frac{S\left( {1 + a} \right)}{\left( {1 - a} \right)}}.} & (1.88)\end{matrix}$

As can be seen, a real solution to (1.88) is only possible if a<1 andcan be given as,

$\begin{matrix}{t_{OPT} = {{\sqrt{S}\sqrt{\frac{\left( {1 + a} \right)}{\left( {1 - a} \right)}}{for}a} < 1.}} & (1.89)\end{matrix}$

The direct result of (1.89) gives,

$\begin{matrix}{t_{OPT} = {b\sqrt{S}}} & (1.9)\end{matrix}$${{where}b} = {\sqrt{\frac{\left( {1 + a} \right)}{\left( {1 - a} \right)}} > 1}$fora < 1

which, proves our case without even solving the non-linear equation(1.77) at all! Since b>1 for a<1,

t _(OPT)>√{square root over (S)}  (1.91)

It also states that since a is a function of frequency ƒ, length l,which defines the desired inductance value L and resistivity ρ. As aresult, the optimal thickness t_(OPT) is a function of these threeadditional parameters, not a just a function of t_(OPT0)=S^(0.5) asderived earlier for the constant current density assumption. It also hasto be noted that (1.90) gives the same result for a=0 which correspondsto zero frequency and for this case b=1 and satisfies thet_(OPT0)=S^(0.5) result!

Since the cross-sectional area is a given value of S, for t_(OPT) we canalso define the optimal width w_(OPT) that satisfies,

S=w _(OPT) ·t _(OPT)  (1.92)

giving,

$\begin{matrix}{w_{OPT} = {\frac{S}{b\sqrt{S}} = {{\frac{\sqrt{S}}{b}{where}b} > 1.}}} & (1.93)\end{matrix}$

As a result of (1.90) and (1.93), the high-Q inductor rectangularcross-section has to be a high aspect ratio cross-section having largerthickness than width, or in other words t>w. This is one of the keyinsights of this invention.

The aspect ratio Δ_(OPT)=t/w can be defined as,

$\begin{matrix}{\Delta_{OPT} = {\frac{t_{OPT}}{w_{OPT}} = {\frac{b\sqrt{S}}{\frac{\sqrt{S}}{b}} = {b^{2} > 1.}}}} & (1.94)\end{matrix}$

The same fact could have been proven by writing (1.79) as a function ofw instead of t giving,

$\begin{matrix}{{{\frac{\left\lbrack {1 - {\left( {1 + v} \right) \cdot e^{- v}}} \right\rbrack}{v}\ln\left( \frac{2lw}{w^{2} + S} \right)} + \frac{\left( {w^{2} - S} \right)}{\left( {w^{2} + S} \right)}} = 0.} & (1.95)\end{matrix}$

As can be seen, equation (1.95) is a non-linear function of w. In thiscase (1.84) becomes,

$\begin{matrix}{{a + \frac{\left( {w^{2} - S} \right)}{\left( {w^{2} + S} \right)}} = 0.} & (1.96)\end{matrix}$

The solution this time leads to the solution for w of,

$\begin{matrix}{w^{2} = {S{\frac{\left( {1 - a} \right)}{\left( {a + 1} \right)}.}}} & (1.97)\end{matrix}$

giving,

$\begin{matrix}{w_{OPT} = {{{\sqrt{S} \cdot \sqrt{\frac{\left( {1 - a} \right)}{\left( {a + 1} \right)}}}{for}a} < 1}} & (1.98)\end{matrix}$

which (1.98) gives the identical result of (1.93) satisfyingS=w_(OPT)·t_(OPT). Furthermore it can also be proven that W_(OPT)<δ, orin other words the shorter dimension of the rectangular cross-sectionswill be less than the skin depth δ for all of the typical inductorvalues. Thus (1.97) gives the identical result of (1.93).

As a conclusion of this analysis we can state:

The Q_(MAX) for a rectangular cross-section by enforcing the constantarea constraint S is a rectangular cross-section having a largerdimension along z axes, which is the thickness t, compared to its width(w). It is NOT a square. Furthermore, there is an optimal thicknesst_(OPT) that can be found exactly by solving (1.79) for any givenfrequency, inductor value and cross-sectional area of the spiralinductor windings. The exact thickness t_(OPT) which is the solution of(1.79) has a weak dependency to resistivity, frequency of operation andthe desired inductance value L due to the l dependency in (1.77), but itis a function of these parameters and does not appear in the previouslyderived t_(OPT0).

This conclusion is verified in FIG. 1.14 .1 to FIG. 1.17 .2 which showthe Q versus Al metal thickness t, varying between 0.1 to 500μ, for 20,40, 60 and 80 nH inductors for constant area S constraints of 50, 250,500 and 1,000μ² for 2 different current boundary conditions mentionedearlier along with the constant current density approximation at afrequency ƒ at 100 MHz.

The top set of curves in each plot of these figures corresponds to theuniform current density assumption. As can be seen, the Q peaks arecorresponding to thicknesses of the square root of S at 7.07μ, 15.81μ,22.36μ and 31.62μ for S=50, 250, 500 and 1,000μ², respectively, asderived earlier. These solutions are marked with a vertical dotted linein each plot. As can be seen, the thickness for the Q peaks areindependent of the L values, but the peak Q value is a function of thedesired inductance values taken as 20, 40, 60 and 80 nH as predictedanalytically.

The case of forced uniform current density at the surface of theconductor gives two peaks, one below and the other one after t_(OPT0).It is noteworthy that in this case there is a double peak with a minimumat t_(OPT0). The reason for a double peak is that a scan of thethickness from 0.1μ to 500μ till t<S^(0.5) yields a width w that ishigher than the thickness t, to satisfy the constant area S conditionimposed, and after t>S^(0.5) the thickness t becomes larger than thewidth w. Forcing a uniform current density boundary condition at thesurface of the conductor gives symmetric results with respect toS^(0.5).

In the case where constant current density along the thickness (along zaxes) is assumed, the curves start with very low Q's for smallthicknesses. This is due to the fact that for very small thickness t,will give very large widths, w to satisfy the constant area S condition.All the curves have single peaks, which matches the second peak in theearlier case, because as the thickness increases the surface currentdensity at the surface of the conductor case applicable to the metalgeometry. The optimal thicknesses t_(OPT) where the QPEAK for all casesfor S=50, 250, 500 and 1,000μ², at 100 MHz are significantly higher thant_(OPT0) as predicted by the earlier analysis. However, as can be seen,none of these metal thicknesses shown in FIG. 1.14 .1 to FIG. 1.17 .2can be provided by any known IC process. This limitation is addressedherein below.

FIG. 1.18 .1 show the magnitude of QPEAK variation as a function offrequencies for the inductors 20, 40, 60 and 80 nH for 100 and 200 MHz.As can be seen, the peaks are at the same t_(TOPT0) giving linearlyscaling peak values with frequency as predicted. FIG. 1.18 .2 show theuniform current density and current density distribution with “singlesided solution” approximation together to show the relative differencesbetween two assumptions. As can be seen, the Q(t) variations in the“single sided solution” approximation gives much more realistic results.To have a better representation of the realistic “single sided solution”approximation they are plotted alone in FIG. 1.19 .

The “Complete Solution Current Density Distribution” for Spiral Windings

In spiral inductor windings the current density distribution in thewindings is different than the straight wire with a rectangularcross-section. The magnetic fields in the windings are also a functionof the distance of the winding from the center (air core) of the spiralinductor, and also vary along the winding and its z coordinate. One cansee this very easily by applying Ampere's law to the spiralcross-section. The magnetic field distribution in the winding is thecause of the all the combined eddy currents, skin and proximity effects.The solution requires three dimensional numerical simulation of theentire spiral inductor. The normalized magnitudes of the magnetic fieldin the windings for widths of 0.5δ, δ, 2δ, 3δ, 4δ, 5δ and 10δ are shownin FIG. 1.20 .1 for a spiral inductor having four windings, assumingthat each winding is carrying the same current I. As can be seen, the Bfield enters from both sides of the windings, with the exception of theinner most winding and will create similar current density distributionsin the windings. The innermost winding normalized B-field distributionis shown in FIG. 1.20 .2 and it looks very similar to the solutionobtained with “single sided solution”. Looking at the normalized B-fielddistributions in FIG. 1.20 .1 one can also see that these distributionssatisfy Ampere's law. FIG. 1.21 .1 show how R_(AC)/R_(DC) compare to the“single sided” and “complete solution” results, the x axis beingwinding-width normalized with the skin depth, w/δ. FIG. 1.21 .2 showR_(AC)/R_(DC) comparing the “single sided” and “complete solution”results, the x axis being the winding width w. One interesting propertythat is fairly clear in FIG. 1.21 .1 and FIG. 1.21 .2 is that assuming a“single sided solution,” estimates R_(AC)/R_(DC) can be corrected basedon a “complete solution.”

The short summary of the curves related to QPEAK vs t_(OPT) at 100 MHzfor different cross-section areas and inductor values is given in Table[2]. More detailed explanations and critical parameters for differentinductor values of 20, 40, 60 and 80 nH at 100 and 200 MHz are shown inTables [2.1-2.4] below.

TABLE 2.1 f = 100 MHz f = 200 MHz t_(OPT) (μ) 195.9 264 w_(OPT) (μ)[S/t_(OPT)] 2.552 1.894 Q_(max) 11.26 10.69 Δ(t_(OPT)/w_(OPT)) 76.77139.4 Δ_(δ) (w_(OPT)/δ) 0.307 0.322 Length (μ) 17,600 18,400 R_(DC) (Ω)0.91612 1.005 R_(AC) (Ω) 1.116 1.175 L = 20 nH, S = 500μ², w_(OPT0) =t_(OPT0) = S^(1/2) = 22.36μ, Length = 14,360μ δ = 8.316μ@100 MHz, δ =5.880μ@200 MHz R_(DC) = 0.7841 Ω, R_(AC)@100 MHz = 2.345 Ω, R_(AC)@200MHz = 3.176 Ω Δ(t_(OPT0)/w) = 1, Δ_(δ)(w_(OPT0)/δ) = 2.693@100 MHz,Δ_(δ)(w_(OPT0)/δ) = 3.803@200 MHz

TABLE 2.2 f = 100 MHz f = 200 MHz t_(OPT) (μ) 213.3 287.4 w_(OPT) (μ)[S/t_(OPT)] 2.334 1.74 Q_(max) 12.44 11.86 Δ(t_(OPT)/w_(OPT)) 91.03165.2 Δ_(δ) (w/δ) 0.2818 0.2958 Length (μ) 32,250 33,610 R_(DC) (Ω)1.761 1.835 R_(AC) (Ω) 2.021 2.12 L = 40 nH, S = 500μ², w_(OPT0) =t_(OPT0) = S^(1/2) = 22.36μ, Length = 26,410μ δ = 8.316μ@100 MHz, δ =5.880μ@200 MHz R_(DC) = 1.442Ω, R_(AC)@100 MHz = 4.312 Ω, R_(AC)@200 MHz= 5.841 Ω Δ(t_(OPT0)/w_(OPT0)) = 1, Δ_(δ)(w_(OPT0)/δ) = 2.693@100 MHz,Δ_(δ)(w_(OPT0)/δ) = 3.803@200 MHz

TABLE 2.3 f = 100 MHz f = 200 MHz t_(OPT) (μ) 222.6 299.9 w_(OPT) (μ)[S/t_(OPT)] 2.246 1.667 Q_(max) 13.14 12.55 Δ(t_(OPT)/w_(OPT)) 99.12179.9 Δ_(δ) (w_(OPT)/δ) 0.2701 0.2835 Length (μ) 46,040 47,910 R_(DC)(Ω) 2.514 2.616 R_(AC) (Ω) 2.869 3.004 L = 60 nH, S = 500μ², w_(OPT0) =t_(OPT0) = S^(1/2) = 22.36μ, Length = 27.820μ δ = 8.316μ@100 MHz, δ =5.880μ@200 MHz R_(DC) = 2.065Ω R_(AC)@100 MHz = 6.175Ω, R_(AC)@200 MHz =8.365 Ω Δ(t_(OPT0)/w_(OPT0)) = 1, Δ_(δ)(w_(OPT0)/δ) = 2.693@100 MHz,Δ_(δ)(w_(OPT0)/δ) = 3.803@200 MHz

TABLE 2.4 f = 100 MHz f = 200 MHz t_(OPT) (μ) 222.6 313 w_(OPT) (μ)[S/t_(OPT)] 2.246 1.598 Q_(max) 13.65 13.05 Δ(t_(OPT)/w_(OPT)) 99.12195.9 Δ_(δ) (w_(OPT)/δ) 0.2701 0.2717 Length (μ) 59,120 61,780 R_(DC)(Ω) 3.228 3.373 R_(AC) (Ω) 3.684 3.852 L = 80 nH, S = 500μ², w_(OPT0) =t_(OPT0) = S^(1/2) = 22.36μ, Length = 48,850μ δ = 8.316μ@100 MHz, δ =5.880μ@200 MHz R_(DC) = 2.667Ω R_(AC)@100 MHz = 7.977Ω, R_(AC)@200 MHz =10.80Ω Δ(t_(OPT0)/w_(OPT0)) = 1, Δ_(δ)(w_(OPT0)/δ) = 2.693@100 MHz,Δ_(δ)(w_(OPT)/δ) = 3.803@200 MHz

Performing the analysis at any frequency gives similar results,indicating that a thick metal with widths in the order of the skin depthis needed to achieve a high-Q straight inductor. Spiral inductors can bethought of as straight lines which are coupled with desirable sign andhighest possible mutual inductances. The following section is related topointing out the inherent advantage of spacing the high aspect ratiothick metals, again with high aspect ratio spacing rules for making veryhigh performance spiral inductors.

Since thick metal processes on the order of metal thicknesses shown inFIG. 1.14 .1-FIG. 1.17 .2 and summarized in Tables [2.1-2.4] are notavailable in standard IC processes, one needs to explore processes withmetal thickness of this order that can be realized, along with metalwidths giving high aspect ratios—in other words, widths on the order ofskin depth dimensions with comparable metal-to-metal spacing processingcapabilities. The initial results suggest that if there is a high-Qinductor requirement, these inductors must be designed and built using adifferent process, generally resulting what are called interposerstructures, built with IC processing techniques and connected to atraditional IC with 3D stacking methods. On the other hand, for lowcurrent density (small S) applications, if the IC process allows 4μmetal thicknesses with 2.8μ width and spacing, giving a metal aspectratio of Δ_(M) (t/w=1.43) that type of IC process might be enough formany RFIC inductors for ƒ>900 MHz applications. Known IC designs using4μ metal thickness do not employ the high aspect ratio rule as given inthis disclosure, however. Before proposing an interposer processrequirement, one needs to also explore the geometrical size of thesehigh value inductors to find out if they are within feasible dimensionsthat can be connected to a conventional integrated circuit. Hence, theforegoing proofs and formulae become useful analytic tools.

Inner Space Dimension Optimization of the Spiral Inductor

Spiral air-core inductors have an inner space where there are nowindings. The geometry of inner space can be almost anything, but inmost practical applications they are circular, rectangular, square oroctagonal areas. Among these four most commonly seen inner spacegeometries, probably the square inner space geometry is the most populargeometry due to many reasons, such as better packing density, ease ofdesign and better performance compared to rectangular geometry from anydesign specification. The most critical part in area reduction in spiralinductor having a constant spacing s between windings is winding width wand the inner dimension d_(IN), where the spiral inductor windings startand winds outward from this square shaped inner geometry. Having thickmetal and using the high aspect ratio metal rule as herein taughtminimizes the spiral area very significantly and improves high frequencyperformance as shown above. Similar optimization is needed foroptimizing the inner dimension of a square inner space, which can beextended to any type of inner space geometry. The present inventionreveals that d_(IN) optimization should have a minimum inner dimensiongreater than a mathematically defined d_(INMIN).

To make highly efficient high coupling metal structures, one needs toconsider also the Q value of the individual structures, such as eachleg, besides their highest possible mutual inductive coupling ratioswith other legs, as explained above. The use of high coupling ratio legswith individually low Q structures increases the loss and does notcontribute to the inductance value at elevated operating frequencies.The analytical expression for Q_(LRAC) (1.62) shows that Q_(LRAC) (l, S,t, ƒ) has a logarithmic length dependency. This is clearly shown in theQ_(LRAC) (l, S, t, ƒ) plot in FIG. 1.22 for S=50, 250, 500 and 1,000μ²,at 100 MHz for Al metal width of w=2μ.

According to the invention there should be no inductive structure, suchas any leg in a spiral inductor, having its individual Q_(LRAC)<1 forthe operating frequency of the spiral inductor. The smallest leg lengthin a spiral inductor is determined by the inner dimension d_(IN) asshown in FIG. 1.6 .1 in a square spiral. The d_(IN) is an importantspiral inductor design parameter that greatly affects the overall Q ofthe spiral inductor. According to the invention d_(IN) is calculatedsuch that Q(d_(INMIN), S, t, ƒ)>1 from the analytical expression (1.62).(By contrast, most known spiral inductors have a very small innerdimension, teaching away from the present invention, but which increasesthe loss and reduces the Q of the spiral inductor, a characteristicevidently not recognized in the prior art as a factor in classical booksin the field [31]. In practice there are properly designed inductorswhich might have a larger inner dimension, but there is no rule thatdetermines the selected inner dimension of any inner space geometry. Inthe past, the selection was based on experimental measurement datacollected from many spiral inductor designs for that particular processwithout knowing the reason.)

According to this invention, a minimum inner dimension d_(INMIN) valueis calculated with the analytical relation (1.62) having lengthssupporting Q(d_(INMIN), S, t, ƒ)>1 for a selected cross-sectional area Sand thickness t at the operating frequency. As can be seen in FIG. 1.22the bottom curve, which corresponds to S=50μ² should have a lengthlarger than 800μ to meet this condition at 100 MHz for Al metal width ofw=2μ. The remaining l_(MIN) values for 5=250, 500 and 1,000μ² correspondto the l_(MIN) of each case having Q(l_(MIN), S, t, ƒ)>1 are met atsmaller lengths of 60-80μ.

Another advantage of a larger inner dimension d_(IN) is seen in thereduction of mutual inductive coupling between inductor legs on bothsides of the inner space of the spiral inductor. Since these inductorshave coupling ratios of opposite signs, minimizing the coupling ratiowill increase the inductance L and therefore increase the Q of thespiral inductor. Mathematically stating, the s<<d_(IN) condition mustalso be satisfied! On the other hand increasing the inductive couplingbetween inductor legs on the same side of the inner empty spaceincreases the inductance value L, thereby increasing the Q of theinductor. Since one generally is interested designing a specific L, onewill achieve this design goal with shorter wire length, giving lessresistance and capacitance resulting in higher Q.

Increase of Mutual Inductance by Using High Aspect Ratio Metal GivingHigher Q and Smaller Area for a Desired Inductance Value

It has now been mathematically proven that high aspect ratio rectangularcross-sectional geometry in the z dimension, facing their largerdimensional sides to each other, yields a definitive advantage in givinghigh Q spiral inductors, due to giving smaller R_(AC) at any frequencyof operation compared to placement as in prior art placement with thesmaller dimension facing each other. Having rectangular cross-sectionconductor windings facing their larger dimensional sides to each otherunexpectedly yields much higher mutual inductive coupling compared tofacing the smaller sides together for the same spacing s. Having highinductive coupling between the legs increases the inductor value of thespiral inductor reducing the total length of the winding for a desiredinductance value L and consequently reducing the resistance and givinghigher Q. This fact is due to the geometric mean distance (g.m.d)reduction of the geometries if arranged as herein disclosed. The legstructures for spiral inductors according to the invention are shown inFIG. 1.24 and FIG. 1.25 as perspective and cross-sectional plots differvery clearly from the structures shown in FIG. 1.9 .1 and FIG. 1.9 .2,having the same cross-sectional winding areas as shown earlier. FIG.1.25 is the mathematical matrix equivalent of the geometry shown in FIG.1.25 . This arrangement was previously shown in FIGS. 1.7 .1, 1.81 and1.9.2 where the non-uniform current distribution analysis was performedwhich showed a significant advantage on the reduction of R_(AC) by themetal arrangement configuration as herein explained. As a result, howone configures the rectangles together, in other words the orientationof metal windings, greatly affects the inductor performance and thetotal area for a spiral inductance having the same inductance value L.

Total Inductance Calculation of a Spiral Inductor Using Circuit Theory(Partial Inductance)

An easier way of explaining the total inductance calculation of a spiralinductor from its metal geometry is through circuit theory, rather thanelectromagnetics theory through the “Partial Inductance” concept [1,10-24]. FIG. 1.23 shows the cross-sectional geometry obtained by slicingthe top view of a square spiral inductor along the marked “y cut” plane.As can be seen, the metal aspect ratio Δ_(M), is a large numberaccording to the invention, but the formulation given here applies forany cross-sectional metal winding geometry. The three-dimensionalperspective view is also shown in FIG. 1.24 for adjacent three windingsin one side of the “inner empty area” of the spiral inductor, where thewindings are separated by the inner dimension of the spiral inductornoted as d_(IN). FIG. 1.25 also shows the leg numbering for theinductance matrix generation. The p, C, B notation on the top of FIG.1.25 are the Grover [1] notations for mutual inductance calculationtables and their equivalent variables used in this work are shown witharrows as w, s and t.

We can represent the width and spacing of windings as shown in FIG. 1.26(which is extracted from an analysis of Grover (Ref. [1]) p. 19,Table[1], in terms of metal thickness t and aspect ratios Δ_(M) andΔ_(S) that can be calculated from the notations in this figure.

It is a good practice to have a quick method of estimating the number ofturns needed to design for an inductor value L for a given t, w and srule having the d_(IN) as a parameter, assuming that each side will havethe same number of turns. Although the final structure of a computeddesigned will end up different, this assumption will give a good initialapproximation of the spiral size and idea of the inductance matrixterms, which is very important.

As can be seen the distance d, in the mutual inductances formulationgiven in (1.12) will be replaced by the geometric mean distances (g.m.d)between the cross-sectional metal windings. For any cross-sectionalgeometry and the numbering given in FIG. 1.25 the mutual inductancesgiven in the inductance matrix and their signs will be as,

$\begin{matrix}\begin{matrix}\begin{bmatrix}{L_{11}L_{12}L_{13}L_{14}L_{15}L_{16}} \\{L_{21}L_{22}L_{23}L_{24}L_{25}L_{26}} \\{L_{31}L_{32}L_{33}L_{34}L_{35}L_{36}} \\{L_{41}L_{42}L_{43}L_{44}L_{45}L_{46}} \\{L_{51}L_{52}L_{53}L_{54}L_{55}L_{56}} \\{L_{61}L_{62}L_{63}L_{64}L_{65}L_{66}}\end{bmatrix} & \begin{bmatrix}{+ {+ +}} & {- {- -}} \\{+ {+ +}} & {- {- -}} \\{+ {+ +}} & {- {- -}} \\{- {- -}} & {+ {+ +}} \\{- {- -}} & {+ {+ +}} \\{- {- -}} & {+ {+ +}}\end{bmatrix}\end{matrix} & (1.99)\end{matrix}$

The negative signs in the inductance matrix (1.99), which are for theleg inductors in the opposite sides of the inner space, originate fromthe current direction going in or out of the reference bubble placementon each mutually coupled inductor of the spiral inductor. In any casethe inductance matrix will be a symmetric matrix. Since the leg width wand spacing between the legs s in each side are kept constant andd_(IN)>s, the following relations between the magnitudes inductancematrix elements also holds for any winding aspect ratio having the samenumbering as given in FIG. 1.25 ,

$\begin{matrix}\begin{matrix}{L_{1,1} > {❘L_{1,2}❘} > {❘L_{1,3}❘} > {❘L_{1,4}❘} > {❘L_{1,5}❘} > {❘L_{1,6}❘}} \\{{L_{2,2} > {❘L_{2,1}❘}} = {{❘L_{2,3}❘} > {❘L_{2,4}❘} > {❘L_{2,5}❘} > {❘L_{2,6}❘}}} \\{L_{3,3} > {❘L_{3,2}❘} > {❘L_{3,1}❘} > {❘L_{3,4}❘} > {❘L_{3,5}❘} > {❘L_{3,6}❘}} \\{L_{4,4} > {❘L_{4,5}❘} > {❘L_{4,6}❘} > {❘L_{4,1}❘} > {❘L_{4,2}❘} > {❘L_{4,3}❘}} \\{{L_{5,5} > {❘L_{5,4}❘}} = {{❘L_{5,6}❘} > {❘L_{5,1}❘} > {❘L_{5,2}❘} > {❘L_{5,3}❘}}} \\{L_{6,6} > {❘L_{6,5}❘} > {❘L_{6,4}❘} > {❘L_{6,1}❘} > {❘L_{6,2}❘} > {❘L_{6,3}❘}} \\{and} \\{L_{i,j} > 0} \\{L_{i,j} = L_{j,i}}\end{matrix} & (1.1)\end{matrix}$

Since the length of each leg (l) is greater than the inner lengths, onecan also indicate that,

l ₃ =l ₆ >l ₂ =l ₅ >l ₁ =l ₄ =d _(IN) L _(6,6) =L _(3,3) >L _(2,2) =L_(5,5) >L _(1,1) =L _(4,4)  (1.101)

The passivity requirement as given in (1.14) also holds between thediagonal and off-diagonals of the inductance matrix having couplingratio K<1 between the diagonals and off-diagonals as stated before. Foran equal length assumption, this will result in diagonal entries of theinductance matrix being always larger than any of the magnitude of theoff-diagonals.

Identical relations can be given for the “x cut” of the spiral inductorin FIG. 1.23 , which have zero mutual couplings between the “y cut”matrix terms due to the fact that “x cut” and “y cut” elements areperpendicular and have zero inductive coupling as shown by the Neumannmutual inductance formula given in (1.11) for a rectangular inner hole.As a result, the complete spiral inductor inductance matrix for the“equal number of legs in each side” assumption can be written with theuse of sub-matrix notation as,

$\begin{matrix}{{L\left( {nxn} \right)}_{SPIRAL} = {\left. \begin{bmatrix}L_{xcut} & 0 \\0 & L_{ycut}\end{bmatrix}\rightarrow n \right. = 12}} & (1.102)\end{matrix}$

where the L_(xcut) and L_(ycut) are (6×6) sub-matrices having propertiesas given in (1.100)-(1.101).

The total effective inductance of the spiral inductor under theassumption of having the same current passing through each leg of itthen becomes the sum of all terms of the (1.102), which is a (12×12)matrix. Since L_(xcut) and L_(ycut) (6×6) are identical sub-matrices,the whole operation for calculating the effective spiral inductance of aspiral inductor having three complete turns can be reduced to,

$\begin{matrix}{L_{SPIRAL} = {2 \cdot {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}L_{i,j}}}}} & (1.103)\end{matrix}$

where n=6 and L_(ij)'s are the L_(xcut) or L_(ycut) are the (6×6)sub-matrix terms in (1.103). To increase the inductance of the spiralinductor then one must increase the positive coupling and reduce thenegative coupling of the inductance matrix as given in (1.103) which isbasically with properties given in (1.100) and (1.101) for a 3 turnspiral inductor. The negative couplings can be reduced as one increasethe d_(IN) and the positive couplings can be increased by reducing thespacing s, between the windings.

The geometric mean distance (g.m.d) between two physical rectangles,which has w>>t, increases with width w and is a weak function of spacings, even for zero spacing s between them W. Therefore putting wide andthin rectangles closer does not increase the mutual inductive couplingbetween them significantly, which is not very obvious. This isillustrated in FIG. 1.29 where (g.m.d) of infinitely thin rectangles(w=0) with spacing of 0.5, 1, 5, 10, 40 and 100μ are plotted versusmetal width ranging from 1μ to 500μ. As can be seen the mutualinductance after 10μ of width becomes almost independent of theirspacing and becomes very close to a linear function of the width w, notthe spacing s W. This property works against the objective, when one hasto make wider metals to increase Q, and it also increases (g.m.d)between them and as a result reduces the mutual inductances between legsseparated by the distance s. The spiral inductors in use today fall intothis “small inductive coupling” category.

The reason for selecting (for analysis) infinitely thin rectangleshaving w=0 is to show the (g.m.d) versus thickness relation to bereconstructed very easily by anyone by the use of tables given in [1].The plot of the related table for calculating g.m.d as given in [1] isgiven in FIG. 1.26 .2, based on parameters shown in FIG. 1.26 .1 withthe exact notation given. FIG. 1.27 is a graph that illustrates t vs. dvalues for an inductor and FIG. 1.28 is a combined plot.

The insight is reducing the geometric mean distance (g.m.d) betweenrectangular cross-sections of the windings by arranging them such thatthey have high aspect ratios and facing larger dimensions toward eachother, as according to this invention. As shown in [1] the (g.m.d)versus thickness for high aspect ratio rectangles arranged as said willbe a function of the aspect ratio of the metals themselves and weaklydependent to the spacing s between them. This is illustrated in FIG.1.30 where the FIG. 1.29 data is superimposed to show the significantdifferences very clearly. The data is shown in FIG. 1.29 is marked byloops at both ends in FIG. 1.30 . As can be seen the geometric distance(g.m.d) between the infinitely thin two rectangles placed according tothis invention for spacing of 0.5 and 1μ gives much smaller (g.m.d)compared to prior-art placement for thicknesses varying from 0.5 to500μ. That would result in higher mutual inductances between them. The(g.m.d) versus thickness curves for even very large spacing range of 5,10, 40 and 100μ starts flat and, larger than the prior-art arrangement,but as the thickness becomes on the order of spacing, it still givesmuch smaller (g.m.d) compared to prior-art placement. (g.m.d) versusthickness but still increases with thickness, and remains an order ofmagnitude less compared to prior-art placement. This property allows forlarge cross-sectional area rectangles to have large mutual inductancesbetween them by placing them with the preferred orientation according tothis invention. The mutual inductance along with very high inductivecoupling K values that are obtained with according to this invention istheoretically and practically impossible following the prior-artplacement protocols for spiral windings.

In summary, the advantage of placement according to the invention liesin optimizing mutual inductance and self inductance. One needs to plotthe mutual and self inductances as a function of thickness. The (g.m.d)versus thickness plot explains the reason for the increase in mutualinductance that is shown; therefore it is important to have thempresented before.

FIG. 1.31 .1 shows mutual and self inductances as a function ofthickness for a 2μ wide 4,000μ long rectangular cross-section structurespiral winding separated by again 0.5, 1, 5, 10, 40 and 100μ compared toprior-structure spaced 0.5p apart and having thickness of 2μ plotted asa function of width. As can be seen both arrangements have the samecross-sectional areas equal to each other as in FIG. 1.31 .1. Asexpected, both the self and mutual inductances decrease as the thicknessincreases, but the mutual inductance remains much higher for 0.5, 1, 5,10, 40 and 100μ spacing compared to the prior-art placement. FIG. 1.31.2 is the same data but shown having thickness in logarithmic scale toshow the small thickness dependency better.

As a result the proposed metallization process in this invention is HighAspect Ratio Metallization along with High Aspect Ratio Metal Spacing,“HARMS.” This results in a “Tight Coupling Condition” between thewindings which can be expressed only with an aspect ratio of metal Δ_(M)and the adjacent winding aspect ratio between the spacing Δ_(S)independent of metal thickness!

“Tight Coupling Condition”

The definition of “Tight Coupling Condition” as herein given accordingto the invention is such that not only the adjacent legs are tightlycoupled with large coupling coefficients K, but also, where there ismore than two legs or segments of windings, a number of the next closestlegs are also coupled strongly with large coupling coefficients K. Theadvantage of this is clear from the inductance matrix and the totalinductance formulation given in (1.99-1.103). Since the effectiveinductance of the spiral inductor is the sum of all of the inductancesin the inductance matrix, increasing their coupling will increase theinductance value. Due to the passivity requirement, one can neverincrease the maximum mutual inductance value between any legs more thanthe value as given in (1.15).

FIG. 1.31 .1 and its logarithmic plot of FIG. 1.31 .2 show self andmutual inductances. Another way of looking at this is by inspecting themutual inductance coupling ratios K, that has been shown in FIG. 1.32which never exceeds 1, no matter how close the adjacent windings areplaced. The only way one can achieve a significant increase in theeffective inductance of the spiral inductor is by increasing the mutualinductance between a number of adjacent legs of each side of the spiralinductor. This can be achieved only if the aspect ratio of the spacingbetween the adjacent windings Δ_(S) remains a large value, on the orderof 5 to 10 along with a winding metal aspect ratio Δ_(M) of 10. Thistight coupling condition can be expressed independent of the metalthickness and with only two target processing metallization parameters,an aspect ratio of metal Δ_(M) and an aspect ratio of the adjacentspacing between windings represented as Δ_(S). A thickness-to-widthratio of at least ten and a width-to-spacing ratio between adjacentturns of at least five attains a tight coupling condition where thecoupling coefficient is well above 0.5 and typically above 0.6 at thesecond third and fourth adjacent windings.

One can represent the width and spacing of windings as shown in FIG.1.33 in terms of metal thickness t and aspect ratios ΔM and Δ_(S) as,

$\begin{matrix}{w = \frac{t}{\Delta_{M}}} & (2.1)\end{matrix}$

and

$\begin{matrix}{S = \frac{t}{\Delta_{S}}} & (2.2)\end{matrix}$

where Δ_(M) and Δ_(S) are both significantly larger than 1. It is clearthat the spacing aspect ratio of winding number 0 to 1 is,

$\begin{matrix}{{\Delta_{S}(1)} = {\frac{t}{S}.}} & (2.3)\end{matrix}$

The spacing aspect ratio of winding number 0 to 2 also becomes afunction of the metal width w of the winding number 1

$\begin{matrix}{{\Delta_{S}(2)} = {\frac{t}{w + {2 \cdot s}}.}} & (2.4)\end{matrix}$

One point clearly evident from (2.4) is that the Δ_(S)(2) can never belarger than Δ_(M)! Therefore as long as Δ_(M)<1, as in most of the priorart spiral inductors, Δ_(S)(2) will always remain less than 1 for anyspacing s>0 value. Therefore it is proper to call this condition ofhaving Δ_(M)>1, the “necessary condition of tight coupling.” The spacingaspect ratio of winding number 0 to 3 similarly becomes,

$\begin{matrix}{{\Delta_{s}(3)} = \frac{t}{{2 \cdot w} + {3 \cdot s}}} & (2.5)\end{matrix}$

Recursively one can formulate that the spacing aspect ratio of windingnumber 0 to n becomes,

$\begin{matrix}{{{\Delta_{s}(n)} = {{\frac{t}{{\left( {n - 1} \right) \cdot w} + {n \cdot s}}{for}n} = 1}},2,3,{\ldots.}} & (2.6)\end{matrix}$

As can be seen in (2.6) Δ_(S)(n) will get smaller with increasing n andagain it will always be smaller than Δ_(M)/(n-1) regardless how small sis taken! Substituting (2.1) and (2.2) in relation (2.6) can be writtenas,

$\begin{matrix}{{\Delta_{S}(n)} = {\frac{t}{{\left( {n - 1} \right) \cdot \left( \frac{t}{\Delta_{M}} \right)} + {n \cdot \left( \frac{t}{\Delta_{S}} \right)}}.}} & (2.7)\end{matrix}$

Eliminating t from (2.7) gives,

$\begin{matrix}{{\Delta_{S}(n)} = {\frac{1}{\left( \frac{n - 1}{\Delta_{M}} \right) + \left( \frac{n}{\Delta_{S}} \right)}.}} & (2.8)\end{matrix}$

As can be seen, (2.8) is only functions of metallization parametersΔ_(M) and Δ_(S) and can be simplified to,

$\begin{matrix}{{\Delta_{S}(n)} = {\frac{\Delta_{M} \cdot \Delta_{S}}{{\Delta_{S} \cdot \left( {n - 1} \right)} + {n \cdot \Delta_{M}}}.}} & (2.9)\end{matrix}$

The goal then becomes to find which combination of Δ_(M) and Δ_(S)values can give Δ_(S)(n)>1 for n>1. With some simple arithmetic, (2.9)becomes,

$\begin{matrix}{{{\Delta_{S} \cdot \left( {n - 1} \right)} + {n \cdot \Delta_{M}}} = {\frac{\Delta_{M} \cdot \Delta_{S}}{\Delta_{S}(n)}.}} & (2.1)\end{matrix}$

One is interested in representing n which becomes larger than 1 from(2.10) having Δ_(M) and Δ_(S) as independent process parameters that weneed to enforce. This requires the solution of n from (2.10) as,

$\begin{matrix}{{{\Delta_{S} \cdot n} - \Delta_{S} + {n \cdot \Delta_{M}}} = {\frac{\Delta_{M} \cdot \Delta_{S}}{\Delta_{S}(n)}.}} & (2.11)\end{matrix}$

Solving n from (2.11) is fairly straightforward, giving,

$\begin{matrix}{{{n \cdot \left( {\Delta_{S} + \Delta_{M}} \right)} - \Delta_{S}} = \frac{\Delta_{M} \cdot \Delta_{S}}{\Delta_{S}(n)}} & (2.12)\end{matrix}$

and resulting in,

$\begin{matrix}{{n\left( {\Delta_{M},\Delta_{S}} \right)} = {{\frac{\Delta_{S} + \frac{\Delta_{M} \cdot \Delta_{S}}{\Delta_{S}(n)}}{\left( {\Delta_{S} + \Delta_{M}} \right)}{for}{\Delta_{S}(n)}} > 1.}} & (2.13)\end{matrix}$

As pointed out earlier as the “necessary condition of tight coupling” isΔ_(M)>1, it is logical to plot (2.13) as a function of Δ_(M) and as afamily of curves for several Δ_(S). FIG. 1.34 shows the plot of (2.13)as a function of metal aspect ratio Δ_(M) for spacing aspect ratiosΔ_(S)=5, 10 and 15. They axis shows the number of adjacent couplings nwhich will have Δ_(S)(n)>1 satisfying the “tight coupling condition”.Since n has to be an integer, in these family of curves we will pick theclosest integer below the curves. As can be seen with Δ_(M)=10, andΔ_(S)=5, the tight coupling condition Δ_(S)(5)>1 for of n=5 can beachieved. This physically means that every winding is tightly coupleduntil its 5th neighbor to the right and left, in each side of the spiralinductor, maintaining Δ_(S)(n)>1.

FIG. 1.35 is the same plot with more aggressive Δ_(S)(n)>2, targetparameter. As can be seen with Δ_(M)=10 and Δ_(S)=5 the tight couplingcondition Δ_(S)(n)>2 for n=3 can be achieved. This again means thatevery winding is tightly coupled until its third neighbor to the rightand left, in each side of the spiral inductor, maintaining Δ_(S)(n)>2.

The Total Inductance and Q Increase with “Tight Coupling Condition”

Although FIG. 1.34 and FIG. 1.35 show very clearly the geometricadvantage of having high aspect ratio metal and spacing, independent ofthe metal thickness resulting in the “tight coupling condition”, onealso needs to show the increase in the inductance value and Q of aspiral inductor employing this technique. This can be shown for selectedmetal thicknesses again as a function of metal aspect ratio Δ_(M) forsome selected values of metal spacing aspect ratio Δ_(S).

FIG. 1.36 shows total spiral inductance for a 4 turn spiral inductorhaving inner space d_(IN)=500μ, and metal thickness of 50μ as a functionof metal aspect ratio Δ_(M) and for metal spacing aspect ratios Δ_(S)=5,10 and 15. As can be seen, the curves have a higher slope compared toFIG. 1.34 and FIG. 1.35 for metal aspect ratios Δ_(M)<10, but they goflatter after Δ_(M)>10. As can be seen the increase in the total spiralinductance value is very significant by increasing the metal aspectratios up to Δ_(M)=10, but after that the increase in the inductancebecomes not that significant. FIG. 1.37 -FIG. 1.39 show again totalspiral inductance for a 4 turn spiral inductor having inner spaced_(IN)=500μ, and metal thickness of 100, 200 and 300μ. As can be seenfor all the thicknesses the relative increase in the total spiralinductance in the metal aspect ratio range of 2<Δ_(M)<10 is not lessthan 50% for Δ_(S)=5, 10 and 15.

FIG. 1.40 show the metal thickness effects on total spiral inductancefor t=50, 100 and 200μ, for again a four-turn spiral inductor but thistime having inner space d_(IN)=600μ for metal spacing aspect ratios ofΔ_(S)=5, 10 and 15 in the same plot to give a good view of the fullpicture.

FIG. 1.41 show Q vs Δ_(M) for metal thicknesses t=50, 100 and 200μ for100 MHz. As can be seen, for all metal thicknesses there is a Δ_(M)value at a selected frequency of operation where the Q value peaks andthese peaks are all in the vicinity of Δ_(M)=10. Thus, there is littletheoretical advantage in aspect ratios for either Δ_(M) or As greaterthan 10.

The placement of such high aspect ratio spiral inductor windings Δ_(M)along with high aspect ratio adjacent spacing Δ_(S) according to theinvention also results in several other additional important advantagesover prior-art arrangement which need to be mentioned.

i) Area Reduction for Same Number of Turns and Inner Dimension d_(IN)

Area reduction according to the invention is fairly straightforward andit is an evident advantage of the preferred arrangement of windings, butthis has a very practical advantage. It is a geometry-driven advantage.There is no need for electromagnetic analysis. Simply stated, the spiralouter dimension will be smaller if one uses smaller width and spacing.As shown in FIG. 1.23 the outer dimension of the spiral inductor d_(OUT)is related to the winding width d_(W) and inner space dimension d_(IN)as,

d _(OUT) =d _(IN)+2·d _(W).  (2.14)

On the other hand, the winding width d_(W) is related to w and spacing sas,

d _(W) =s·(n−1)+w·n  (2.15)

where n is the number of turns.

Even if there was no increase in the mutual inductive terms between legsas was shown earlier, according to the invention, a very significantreduction in the total spiral area can be achieved as shown in (2.14)and (2.15). As an example, if the cross-sectional area S is keptconstant, for a thick and high aspect ratio metallization process (2.15)becomes,

$\begin{matrix}{d_{W} = {{S \cdot \left( {n - 1} \right)} + {\frac{S}{t} \cdot {n.}}}} & (2.16)\end{matrix}$

Expression (2.15) can also be written in terms of Δ_(M) and Δ_(S) for agiven metal thicknesses t=50, 100, 200 and 300μ for number of turns n=5as a function of Δ_(M) and the results are shown in FIG. 1.42 . FIG.1.43 is basically the FIG. 1.42 with logarithmic value of they axisshowing a better view of the winding width for smaller Δ_(M) values. Ascan be seen Δ_(M)=10 and Δ_(S)=10 and even Δ_(S)=5 combinations givesvery significant winding width savings which translates to small spiralinductor footprint which cannot be achieved with any prior artprocesses.

ii) Dead Area Generation Reduction, Reduction in R, C and Increase in Lfor Same Number of Turns and Inner Dimension d_(IN)

The advantage obtained for this case of dead area reduction is not asstraightforward and evident as in the previous case, but it has asignificant effect on resistance and capacitance reduction along witharea reduction in spiral inductors. FIG. 1.44 and FIG. 1.45 illustratesome critical geometrical parameters related to the “dead area”generation for each turn. Each 90° turn in a rectangular spiral createsan increase in resistance compared to a straight run, as well as addcapacitance and perturb the current distribution for a distance relatedto the width of the leg, and it does not contribute to the inductance. Aconservative value for the region where the current is perturbed alongthe straight path is 0.5 w. Since this perturbed region will be at bothsides of the leg we can define an effective spiral leg length where theleg length l_(eff) which can be assumed inductive as a function of drawnlength l as,

l _(eff) =l−w  (2.17)

Again by using the constant S rule to maintain a desired cross-section,(2.17) becomes,

$\begin{matrix}{l_{eff} = {l - {\frac{S}{t}.}}} & (2.18)\end{matrix}$

Assuming uniform current density along z axes, the additional “deadarea” resistance is,

$\begin{matrix}{R_{TURN} \approx {\frac{{1.5}\rho}{t}.}} & (2.19)\end{matrix}$

Where ρ is the resistivity of the metal winding. This dead area for eachturn can be also given as,

$\begin{matrix}{{S_{DEAD} \approx {2 \cdot w^{2}}} = {2{\frac{S^{2}}{t^{2}}.}}} & (2.2)\end{matrix}$

This “dead area” will produce an undesired capacitance increase for eachturn proportional to (2.20). As we can see in (2.17)-(2.20), accordingto the invention, increasing the thickness t increases the inductiveportion of the spiral, reduces the “dead area” thus giving smallerparasitic capacitance and resistance and all this results in a smallerspiral with higher Q.

iii) Overall Capacitance Reduction

Unfortunately the above effect cannot be shown analytically as easilyand as clearly as has been done for inductance increase and decrease inthe resistance on a per-leg basis. Increasing thickness will alsoincrease capacitance per length. Assuming a parallel plateapproximation, each interior leg will have capacitance as,

$\begin{matrix}{C_{LEG} = {{{2 \cdot \varepsilon_{r} \cdot \varepsilon_{0}}\frac{t \cdot l}{S}} = {2 \cdot \varepsilon_{r} \cdot \varepsilon_{0} \cdot l \cdot {\Delta_{S}.}}}} & (2.21)\end{matrix}$

As can be seen, the capacitance of each leg will increase with increaseof the thickness t or the aspect ratio of the leg-to-leg spacing Δ_(S).Deciding what capacitance per leg is acceptable for an inductor value Lis not analytically obtainable with reasonable and acceptableapproximations. A possible solution is to plot each leg resonancefrequency as a function of space s. By not making s smaller than theself resonance frequency compared to the desired frequency of operationlooks like a solution, but that can be misleading. Therefore it is notpresented in this work.

On the other hand the designer has control on the design parameter s ina rectangular structure which is not generally true in a stackedinductor structure. In stacked inductor cases, metal spacing betweenlayers is determined by the process, which is not in the control of thespiral designer. On the other hand the designer in this thick metallarge aspect ratio interposer structure has control on spacing s, whichoffers very useful design flexibility. However, the analysis must bedone as a full spiral analysis, not a leg by leg basis. Using highaspect ratio metal spacing increases mutual coupling between legs;therefore the spiral inductor value L can be realized in a smaller areaor a smaller number of turns, yielding desired smaller capacitancevalues per leg. The designer can design the spiral with a completeanalysis using PowerSpiral software (a commercially available toolprovided by OEA International of Morgan Hill, Calif.). Therefore, the Qvalue at the frequency of interest can be plotted easily as a functionof spacing, and space optimization can be done on aninductor-by-inductor basis as the last step.

iv) Much Better Inductance vs. Frequency Property

As can be analytically proven [24,25] for circular and as shown earlierfor rectangular cross-sectional rectangular conductor geometries, theR_(AC)/R_(DC) will increase with the square root of the frequency aftera corner frequency ƒ_(C), defined as a ratio of the plane wave skindepth to some geometrical parameters related to the cross-section of theconductor. According to the invention, by making the windings highaspect ratio with the widths not exceeding skin depth one alreadyincreases the corner frequency ƒ_(C) very significantly compared toprior art spiral inductors, but still the R_(AC)/R_(DC) will increasewith the square root of the frequency. Therefore if one merely takesinto consideration only the square root increase of the R_(AC)/R_(DC)with frequency, the Q(ƒ) relation as given in (1.20) becomesproportional to the square root of the frequency instead of being alinear function of frequency!

The internal inductance LINT, of the conductor behaves in a completelyopposite manner; the L_(ACINT)/L_(DCINT) will decrease with the sameratio as the R_(AC)/R_(DC) with the increasing frequency. The Q(ƒ)relation as given in (1.20) then becomes approaching a finite value, notincreasing as a linear function of frequency or proportional to thesquare root of the frequency by only taking the resistance increase withfrequency! Luckily the total inductance has two components being theinternal L_(INT) and external inductance L_(ENT). External inductance isdue to the mutual coupling terms in the overall structure and is nearlyindependent of frequency. Therefore the total inductance does notdecrease continuously as a square root function of frequency; itapproaches to the external inductance value L_(ENT) and the Q(ƒ)relation as given in (1.20) then becomes again proportional to thesquare root of the frequency, which is a very good property as far asbuilding inductors is concerned. According to the invention, byemploying the tight coupling condition, the spiral inductor inherentlyhas much larger external inductance L_(EXT) component and therefore thedecrease of the internal inductance with frequency is reducedsignificantly compared to the prior art spiral inductors. One may definea figure of merit ƒ_(MERIT) as,

$\begin{matrix}{f_{MERIT} = {\frac{L_{EXT}}{L_{TOT}}.}} & (2.22)\end{matrix}$

This ratio in tight coupling condition is a larger value than 0.6meaning that in any case the L_(AC)/L_(DC) ratio cannot be less thanƒ_(MERIT) at any frequency!

v) “Image Spiral” Coupling Reduction

In a typical application of spiral inductors there can be conductiveregions on top or bottom of the spiral inductor such as ground/powerplanes, metallic lid of the package etc. in the z direction. At thesehighly conductive regions the intended spiral inductor can produce eddycurrents and generate loss which is fairly difficult to simulateaccurately. At the extreme case one can assume that these highlyconductive regions acts like infinitely large perfect ground planes andthere will be an “image spiral” formed with respect to these perfectconductive planes but carrying an opposite directional current flow init with respect to the intended spiral inductor [24,25]. A properapproximate equivalent circuit simulation for this assumption will besimulating two coupled identical spiral structures separated by twicethe distance d_(z) to these highly conductive regions, where the inputand output pins of the image spiral are shorted. Higher coupling betweenthe legs of the intended and the image spiral will cause a largerreduction of the effective inductance of the intended spiral inductor.As shown earlier with the tight coupling condition, the inductivecoupling between the windings in the (x,y) direction is kept very high.On the other hand the inductive coupling in the z direction is inverselyproportional to the thickness t of the windings, not the distance d_(z)between them! Having winding thicknesses tin the order of 100-300μ inaccordance with the invention increases the critical distance where theunwanted image spiral effect can become significant. On the other hand,having wide spiral windings with adjacent windings of small thicknessamong each other, as prior art spirals, the z directional inductivecoupling is very large, (parallel rectangular structures placed widersides facing each other) as shown in the tight coupling condition. As aconclusion, the tight coupling condition defined herein—as having Δ_(M)and Δ_(S) on the order of 10 and larger—reduces this image spiral effectvery effectively!

Full PowerSpiral Simulations Showing the Effectiveness of the InterposerMetal Thickness on Spiral Inductor Performance

Until now, the advantage of using a thick metal with large metallizationand spacing aspect ratio for spiral inductor performance has been shownanalytically under certain approximations to give clear understanding ofthe non-obvious issues. In this section is a description of full 3Delectromagnetic simulation results for inductor specifications. Thiswould not even have been thinkable with current metal processing rulesfor IC's, MCM's, PCB or any known advanced packaging technologies. Theinductor values to be designed are taken as 5, 10, 20, 40, 60, 80, 100and 200 nH. Anything above 10 nH is basically considered unrealisticvalues for any known process. To raise the design goal even further, thespecification for DC resistance is kept to very small numbers as neededin FIVR work geared for 100 MHz switching frequencies. The results aregiven in self-explanatory tables in Table [3.1] to Table [6] for metalthicknesses of 50, 100, 200 and 300μ.

TABLE 3.1 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[Ω] 5 42.35 1,000428 0.5 10 40.8 732 438 0.72 20 39.34 424 453 1.06 40 37.51 311 475 1.5760 36.16 259 490 1.96 80 35.75 229 500 2.31 100 34 208 515 2.63 20030.97 131 563 3.94 Thickness = 50μ Space = 1μ, Width = 2μ, S = 100μ²,I_(MAX) = 2 A Δ_(tw) = 25, Δ_(ts) = 50

TABLE 3.2 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[Ω] 5 39.36 1,030444 0.51 10 36.98 608 466 0.76 20 33.49 452 489 1.13 40 30.52 267 5291.69 60 28.74 225 556 2.14 80 26.95 202 583 2.5 100 25.9 164 606 2.92200 22.37 112 687 4.46 Thickness = 50μ Space = 5μ, Width = 2μ, S =100μ², I_(MAX) = 2 A Δ_(tw) = 25, Δ_(ts) = 10

TABLE 3.3 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[Ω] 5 36.69 848457 0.53 10 32.57 639 499 0.8 20 29.07 382 541 1.21 40 24.99 229 5971.83 60 23.41 196 646 2.34 80 221.67 176 681 2.82 100 20.64 144 716 3.24200 17.68 100 835 5.01 Thickness = 50μ, Space = 10μ, Width = 2μ, S =100μ², I_(MAX) = 2 A Δ_(tw) = 25, Δ_(ts) = 5

TABLE 4.1 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[Ω] 5 48 714 4780.3 10 45 474 506 0.64 20 43.4 314 548 0.66 40 38.47 210 611 0.99 6035.31 159 660 1.27 80 33.59 142 695 1.51 100 31.97 123 730 1.73 200 28.280 856 2.67 Thickness = 100μ, Space = 10μ, Width = 4μ, S = 400μ²,I_(MAX) = 8 A Δ_(tw) = 25, Δ_(ts) = 10

TABLE 4.2 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[mΩ] 5 56.2 516497 130 10 54.03 346 542 230 20 51.78 230 596 340 40 50 156 677 520 6049.46 118 731 670 80 46.96 100 785 800 100 45.9 92.6 821 920 200 39.6760.58 983 1430 Thickness = 100μ, Space = 10μ, Width = 8μ, S = 800μ²,I_(MAX) = 16 A Δ_(tw) = 12.5, Δ_(ts) = 10

TABLE 5.1 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[mΩ] 5 65.11 264630 49 10 62 175 702 74 20 59 125 810 111 40 52 85.4 972 170 60 50 69.21080 221 80 45 58.6 1152 262 100 40 50.8 1224 302 200 35 35 1494 469Thickness = 200μ, Space = 20μ, Width = 16μ, S = 3,200μ², I_(MAX) = 32 AΔ_(tw) = 12.5, Δ_(ts) = 10

TABLE 6 L(nH) Q_(MAX) f_(QMAX)[MHz] d_(OUT)[μ] R_(DC)[mΩ] 5 61.74 116920 11 10 53.99 79 1080 17 20 49.77 55 1320 25 40 44 37 1560 39 60 40 30731 51 80 35 25.6 1920 61 100 32 22 2040 70 200 29 15.4 2520 100Thickness = 300μ, Space = 20μ, Width = 60μ, S = 18,000μ², I_(MAX) = 180A Δ_(tw) = 5, Δ_(ts) = 15

All of the simulated metal thickness, spacing and widths are possible inthe existing interposer technology, so these are not fictitiousmetallization rules. As can be seen, the interposer technology withcorrectly suited metal thickness can deliver these heretofore unheard oflarge value inductors in the order of 1 mm by 1 mm size yieldingheretofore unheard of performance, not even reported for small-valueon-chip or in-chip inductors!

As can be seen in Tables[3.1] to Tables[6] for high efficiency, highcurrent buck converter applications for a load current in the order of 2A, the R_(DC)<20 mΩ limit as given by (1.17) can be achieved with 300μof metal thickness and up to 20 nH values giving maximum inductorcurrent of 180 A realized in an approximately 1 mm by 1 mm area. Thismeans that larger load currents can only be achieved by having multipleinductors driven by Poly-Phase architectures [3-9].

The small size of even unheard of large inductors allows many inductorsto be integrated on the interposer and opens up large opportunities inFIVR and RFIC design space and even allows multiple FIVR architecture.This represents a major breakthrough achievable with miniaturizationaccording to the invention.

In reality one cannot come up with a standard IC metallization processhaving any desired large values for Δ_(M) and Δ_(S) as shown anddescribed in connection with FIG. 1.34 -FIG. 1.43 . One must insteadselect a set of practically achievable and good yielding Δ_(M) and Δ_(S)values. In the light of FIG. 1.34 -FIG. 1.43 it is fairly clear that thepractical processing limits of the HARMS process can be set for Δ_(M)=10and Δ_(S)=5-10 and get significantly better size, inductance value and Qimprovement compared to prior art spiral inductors. Even thesecalculated Δ_(M) and Δ_(S) values are not achievable with standard metalprocesses today used in IC technologies. As we mentioned earlier thethickest metal process available today is only at the metal 5, havingthickness of t=4μ and width and spacing w=s=2.8μ giving onlyΔ_(M)=Δ_(S)=1.42 and as can be seen these numbers are at the very lowend of the FIG. 1.34 -FIG. 1.43 not satisfying the derived “tightcoupling” condition. At lower levels of metals, which are all less than1μ thick, with fine lithography one can achieve the desired Δ_(S) valuesneeded for the “tight coupling” condition today, but to meet thecross-sectional area S for desirable DC resistance and/or currentdensity requirements they can only be realized with very wide metalslines, which results in very small Δ_(M)<1 values which again doesn'tsatisfy the “tight coupling” condition as defined in this invention. Asclaimed in this invention, we need both Δ_(M) and Δ_(S) have to be inthe order of 10 and 5-10, respectively, for any thickness of metaldesired to see a significant improvement on spiral inductor area andperformance.

Even if we speculate that one develops an IC metallization thickness inthe order of 20μ with the same etching rule presently giving Δ_(S)=1.42,we still cannot meet the “tight coupling” condition. As a result we needto come up with a metallization etch process than can give the desiredminimums of Δ_(M)=10 and Δ_(S)=5-10, which is probably not possibletoday and for some time in the future. Instead, the present inventionprovides an alternative technique for realizing such structures.

Silicon etch technologies offers such an alternative. Silicon etchtechnologies have improved very significantly over the years[34-35]. Sietch structures with aspect ratios (depth/width), exceeding even 25 havenow been demonstrated. These processes are used in Silicon Through Via(STV) technologies, 3D stacked packaging, Micro-Electro-MechanicalSystems (MEMS) and many more other applications[36-41]. The presentinvention exploits the underlying processes in a whole new field ofapplications, giving what the present inventor has designated as the“HARMS” process. However, HARMS uses the high aspect ratio Si etchcapability similar to Deep Reactive Silicon Etching (also known as Boschprocesses)[33], in a reverse mode of trenching. Similarly, the method ofmetallization which is used in “HARMS” technology is derived from thetotally opposite usage of the STV technology. In STV Si is etched,making a void or a deep trench and the void or the trench is filled withmetal. In “HARMS” technology according to the invention, Si is etchedcreating a “wall” or a “column” herein called a “Si Core,” and the metalencapsulates the “Si Core,” creating the spiral windings and pads whichwill be the spiral inductor electrical connections to the externalcircuits. The encapsulating metal thickness is restrained to on theorder of skin depth k×δ where k=1.6-3.85 as claimed before givingR_(AC)/R_(DC)<2 at the desired frequency of operation of the spiralinductor.

Current HARMS processing guidelines are outlined in Table [7] as shownbelow for several desired “final” dimensions.

TABLE 7 Metal Thickness Minimum Width Minimum Spacing (t) [μ] (w) [μ]t/w = 25 (s) [μ] t/s = 10 50 2 5 100 4 10 200 8 20 300 12 30

“HARMS” (High Aspect Ratio Metallization and Spacing) InterposerTechnology Processing Capabilities

FIG. 2.1 and following illustrate some critical processing steps of theinventive HARMS process in an interposer-based spiral inductor. Such aninductor is mounted to a silicon-wafer-based circuit. There can be threebasic choices of starting materials.

i) As shown in FIG. 2.1 , the starting material can be a conventionalSOI wafer. FELT, Bonded wafer and other types are common choices in theIC industry. In these types of structures the overall structure consistsof two different thicknesses of high-quality single-crystal Si separatedby a fairly thick layer (1-2μ) of SiO₂. The thicker portion of the Si istypically used for mechanical handling, not for active device formation.In the described interposer structure the spiral structures can be builton any one side of the SOI wafer, which can be 50μ-400μ thick, dependingon the desired thickness of spiral inductor windings.

ii) If the desired thickness of spiral inductor windings is greater than200μ, like in many applications shown earlier, there is no need for SOIwafer, a traditional Si wafer with a thermally grown thick (1μ-2μ) layerof SiO₂ will be used as the interposer starting material. As hereinafterexplained in connection with the final result, FIG. 2.32 shows a crosssection of the STV structure connected to both sides with C4 bumps.

Fully integrated IC applications (without interposer). Instead ofbuilding an interposer and connecting it to the IC with C4 bumps, thewhole structure can be formed as one unit. The bottom of the buriedoxide region is of standard wafer thickness, on the order of 300μ-500μ,and is not used for anything other than providing mechanical handlingcapability for the processing steps that follow. Therefore according tothis invention the spiral inductors are built in the “unused” thickunder portion below the buried oxide and connected to the active devicearea by etching holes in the buried oxide and connecting to the firstlayer metallization in the active device area. In both startingmaterials, which can be SOI or SIMOX wafers, the IC will be built withstandard processing steps on one side of the buried oxide, which aredesignated in the following figures as the active layer. The results ofthree different but related processes are illustrated in FIG. 2.26 .4,FIG. 2.26 .11 and FIG. 2.26 .16, which closely resemble the results of astructure made with an interposer as in FIG. 2.26 , as explained below.Shown only is the STV region, as other regions do not differ from aninterposer structure. The starting material can be an SOI wafer asexplained in i) or an SIMOX (oxygen implanted wafer), which is alsocommon in processor ICs or high-performance device processes. FIG. 2.26.1 through FIG. 2.26 .4 are illustrations of the result of processingsteps involved in making fully integrated structures according to theinvention for buried oxide thicknesses between one and two microns andactive IC layer between one and three microns, thus eliminating theinterposer structure. The captions with the figures recite dimensionsand features that would allow one of skill in the art to make thestructure. Similarly, FIG. 2.26 .5 through FIG. 2.26 .11 areillustrations of the result of processing steps involved in making fullyintegrated structures according to the invention for buried oxidethicknesses between one and two microns and active IC layer greater thanthree microns. Finally, FIG. 2.26 .12 through FIG. 2.26 .16 areillustrations of the result of processing steps involved in making fullyintegrated structures according to the invention for a buried oxidethickness of about 20 nm according to the SIMOX process and an activelayer of less than about 200 nm. The processes differ in their abilityto yield consistent metallization through the active layer penetratinginto the substrate material under the buried oxide.

In the SIMOX processes, the active devices are built on top of a verythin silicon layer (typically on the order of 200 nm). The buried oxideis formed by deep oxygen implant and annealed after forming a buriedoxide structures on the order of a thickness of 20 nm. This thickness ofoxide will not act as a etch stop for deep silicon etch, which will bedone in the substrate.

In all three noted processes, after the IC formation process iscomplete, the spiral inductors will be built below the active device andwill be connected by etching the buried oxide and depositing metal fromthe other side connecting it to the previously deposited metal earlierin the active device formation step.

In FIG. 2.2 and hereafter, only the silicon and SiO₂ layers are shown.With FIG. 2.2 as the starting material or Step 1, Step 2 is to provide atantalum deposition of the thickness as noted on the silicon layer. InStep 3 (FIG. 2.4 ) a photoresist layer 5 is deposited in a desiredpattern for tantalum masking representing the structure of the spiralinductor as seen in a plan view (FIG. 0.1 . Step 4 is a standardtantalum etch of minimum 2μ width (FIG. 2.5 ) followed by a Step 5, adeep reactive ion etching of width of 2μ (FIG. 2.6 ). Step 6 is photoresist removal (FIG. 2.7 ) but to show a better idea of the impact ofthis etching, FIG. 2.7 also illustrates the high aspect ratio achievedby a deep etch. In other words, earlier figures are a flattened versionof FIG. 2.7 . In practice, the aspect ratio is actually much moreexaggerated with the aspect ratio closer to 25:1, which is difficult toillustrate. Specific example values are listed with this figure. Thisillustrative figure forms the basis of the following steps.

Step 7 (FIG. 2.8 ) is a second tantalum depositions step in which thesides of the pillars, as well as the valleys and ridges are coated to atypical thickness of 0.5ρ. Step 8 (FIG. 2.9 ) is a directional drytantalum etching step that only etches the valleys and the ridges toremove the 0.5μ deposition layer. A preferred dry etch is a plasma etch,which by its nature is directional. The tantalum portions of thestructure are thereafter electroplated (Step 9, FIG. 2.10 ) to coat thewindings on both opposing surfaces and the top of the ridges. Thestructure is essentially complete, except for interconnections.

FIG. 2.11 and FIG. 2.12 are images of a scanning electron microscope ofa grid structure made in a process based on the invention in a test thatconfirms that such a structure of these deep, narrow and closely spaceddimensions can actually be achieved. In practice, a spiral structurewould be formed with ridges in the shape shown schematically in FIG.2.10 . The shown structures are built with Δ_(Siw)=25 and δ_(TA)=0.5μand t_(Si)=50μ.

FIG. 2.14 is a top plan view of a connector pad as for the IN terminaland OUT terminal of FIG. 2.13 . The width and length dimensions need tobe large enough to accommodate a small solder ball, such as a C4 bump,to connect to an IC or an external pad according conventionaltechnology. The width of the pad must be wider than the winding width toassure effective connectivity. A typical pad W PAD is 75μ to 100μ squarewith a lead tab C_(PAD). Micro-bump technologies permit smaller paddimensions. As explained below, the lead tab may connect to one or moreconductive columns. The cross-section of the columns may be square,rectangular, triangular, circular, oblong, or a combination thereof orinterconnections among combinations. If columns are not used, as forexample shown in FIG. 2.26 , then the C4 bump touching a thick siliconsubstrate simple will not conduct effectively, and in fact cannot evenbe built. FIG. 2.26 .9 through FIG. 2.26 .11 show structures thatprovide effective conduction for C4 bumps on both sides of thestructure.

To achieve a suitable interposer structure with C4 bumps, it isnecessary to embed the inductor and its terminals in a dielectricmaterial. Step 10 (FIG. 2.15 ) is the process of immersion of thestructure in a dielectric, such as by conventional low-pressure chemicalvapor deposition of SiO₂ to be able to select the pad and bumppositioning on the structure. Other alternatives are organic dielectricmaterials with good step coverage, such as polyamide or parylenedeposition to achieve planarization of the structure to more effectivelyconnect with C4 bumps.

Pads in the HARMS process have two main functions and their geometry arepreferably kept as a square with a dimension of W_(PAD), larger thanspiral windings w, regardless of the total width of the spiral windingswas shown in FIG. 2.14 .

The first function of the pad is to provide a low resistivity path fromthe solder bump on top of the interposer to the bottom of the thickspiral windings which is in the order of 300μ, while solder bump rulesfor electrical connectivity is maintained. Their placement and size arebasically dictated by solder bump rules which have pad size which are inthe order of 75-100p placed in a given periodic “pad array spacingrule”. Having pads in an array, rather than arbitrary placement, bringsease in the alignment between the interposer and the bumps on the IC.

Step 11 is the provision of a pad etch for a single pad. Having a padwith a single-square Si pad core as shown in FIG. 2.16 , with a pad sizeof 75-100μ per side gives the desired low resistivity path for manyapplications from a solder bump place on the top of the interposer tothe bottom of the thick spiral windings. The top four curves shown inFIG. 2.34 .2 show the STV resistance as a function of electroplatingthicknesses of δ_(EP)=2, 4, 6 and 8μ as a function of silicon corethickness. As can be seen, the STV resistance and the “contactresistance,” which is the distributed resistance from the top of the C4bump to the bottom of the inductor, can be larger than 2 mΩ for Si corethicknesses exceeding 200μ. As can be seen in FIG. 2.34 .2 the contactresistance can even approach a remarkably small 20 mΩ for δ_(EP)=2μ!Since every inductor will have two C4 bump connection and as shown inFIG. 2.33 the “total resistance” which includes twice the contactresistance plus the active spiral winding resistance of the inductor hasto be kept less than 20 mΩ for DC/DC converter applications. Thereforethis “apparently” low resistance is actually unacceptably high for theseapplications and should be kept less than 1 mΩ!

However, to further reduce resistance to the solder bump on top of theinterposer to the bottom of the thick spiral winding and to create abetter contact, the underlying structure can be built as an array ofsquare Si Columns spaced more closely than the 2δ_(EP) that is typicallyused in array spacing as shown in FIG. 2.17 and FIG. 2.17 .1.Accordingly, Step 11.1 is an alternative step. With this arrangement thespace between the columns will be filled with electroplating metal inthe electroplating step as shown in FIG. 2.17 . This of course issomething that must be completely avoided for spiral windings! Comparedto the simplified Single Si Core pad connection shown in FIG. 2.16 thisarrangement gives much smaller resistance between the solder bump on topof the interposer to the bottom of the thick spiral winding, which canbe shown with 3D resistance simulation. The size of the square Sicolumns is a function of the Si Core thickness and can be calculatedusing the same Δ_(Siw)=25 rule also having a minimum dimension of 2μ by2μ. As an example for Si Core thickness t_(3Si) of 300μ and δ_(EP)=8μ,the square Si column dimension will be on the order of 12μ spaced lessthan 16p. FIG. 2.17 .1 is an example of an arrangement that willsignificantly increase the effective area of the current path from thesolder bump to the bottom of the thick spiral windings and improve yieldfor good contacts.

FIG. 2.34 .1 shows the integer number of silicon columns which can fitin a 100μ pad size as a function of Si Core thickness for electroplatingthicknesses of δ_(EP)=2, 4, 6 and 8μ generating a vertically conductivegrid structure. As can be seen at the far right end of the plots towardsthe 500μ of Si core thicknesses, 3 to 4 Si columns can fit side by sidewith 2δ_(EP) spacing in between them by arranging the Si columndimensions. The lower family of four curves shown in FIG. 2.34 .2 showsthe impact of this Si column array configuration on the STV and “contactresistance” of the spiral inductor compared to the single Si columnarrangement. As can be seen the contact resistance of 1 mΩ thresholdvalue (marked as the bottom dashed line) can be met for this arrangementfor even large Si core thicknesses.

Referring to FIG. 2.18 there is shown the result of a Bump Pad Etch step(Step 12) that opens the dielectric to allow for placement of a C4 bumpconnection to the underlying electroplating metal (EP layer) (Step 13FIG. 2.19 ).

Since electro-plating steps for spiral structures require an electrodeused for electro-plating, all the spiral structures must connect to thiselectrode during the electro-plating process. The wafer-scaleconnections are formed by constructing a grid as in FIG. 2.20 withelectrodes at the periphery. These electrodes are later severed during adicing process whereby the spiral structures are extracted.

As can be seen in FIG. 2.14 showing an individual spiral structure, thepad array rule about regular bump placement forces the spiral structuresto be placed at set locations so that bumps and pads properly align.This pad array rule is satisfied by an accurate and advancedelectromagnetic 3D simulator such as PowerSpiral (available as a productof OEA International of Morgan Hill, Calif.) to automatically design thespiral inductor with a desired inductance value L also meeting itscurrent I, Q, R_(AC), R_(DC) specs by adjusting the width, w and innerdimension, d_(IN) for a given HARMS process rules having a minimal padclearance C_(PAD) and maintaining pad array rules regarding placement ofbumps. The pad array rules can be satisfied easier if both the pads arealigned to a row of pad array and centered relative to the spiralstructure as shown in FIG. 2.14 rather than placing them arbitrarily asshown in FIG. 1.6 .1.

The second function of pads in a structure made according to the HARMSprocess is to create STVs. Making pads as an array of square Si columnswith the spacing smaller than the 2δ_(EP) as shown in FIG. 2.17 willyield STV resistance several times less compared to a simplified SingleSi Core pad connection shown in FIG. 2.16 , and its effect in verticalresistance reduction is shown in detail in FIG. 2.34 .2.

The Scanning Electron Microscope (SEM) photos shown in FIG. 2.11 andFIG. 2.12 display the Si etch capability of holes with finished sizes of10μ by 10μ on a SiO wafer where t_(3Si)=50μ. As can be seen, the Sietched holes have 3μ spacing in between and are very uniformlyencapsulated with finished 0.5μ thick Ta all the way down to SiO₂ layer.The w_(Si) is 2μ, giving Δ_(SiW)=50/2=25 and Δ_(SiS)=50/12=4.16. Theyshow a very uniform conductive grid structure all across a 6-inch wafer.

FIG. 2.27 shows the desired outcome resulting spiral winding dimensionsusing 47.5μ thick “Si Core” with Δ_(SI)=25 rule with 2.0μelectro-plating metallization thickness achieving a tight couplingcondition with Δ_(w)=10 and Δ_(w)=5 with a Si Core to Si Core spacing of15μ. As can be seen, the processing parameters give a very largeelectro-plating metal cross-sectional area S=255μ² which can carry 5.1 Awith J_(MAX)=2×10⁶ A/cm² electro-migration limited current densities.The minimum space between the “finished” windings can be processed withs_(MIN)=10μ giving Δ_(M)=7.14 with a Si Core to Si Core spacing of 15μ.

If there is a need for a larger current and/or lower DC resistance, thenthere are two parameters to adjust as shown in FIG. 2.28 , which are thethickness of the Si Core, t_(3Si) and the electro-plating metalthickness δ_(EP) encapsulating it. Since the resistivity of tantalum(Ta) is approximately 10 times the resistivity of the electro-platingmetal such as copper or aluminum, one may ignore the contribution of thedeposited Ta thickness δ_(Ta) on the resistance to simplify the“back-of-the-envelope” calculation. The structure shown in FIG. 2.28 hasa Si Core t_(3Si) thickness of 291.5μ and δS_(EP)=8μ giving a metalcross-sectional area of S=5,199μ², which can carry 103 A with the sameJ_(MAX)=2×10⁶ A/cm² electro-migration limited current density. As can beseen, both cross-sectional structures meet the “tight coupling”condition disclosed in this description as well as the available HARMSprocessing capabilities. The minimum space between the finished windingscan be processed with s_(MIN)=30μ giving Δ_(M)=10, 47 with a Si Core toSi Core spacing of 77μ. If one wants to achieve the same current or sameresistance per unit length with the thickest metal rules (t=4μ andw=s=2.8μ) available in the IC technology today, one will need to havemetal track widths exceeding 1000μ wide, which is basically unthinkablein any known IC technology! Having these wide metal widths will alsogive virtually no mutual coupling between windings, generating spiralinductors with only internal inductance components and will not even fitwithin any IC dimensions!

FIG. 2.28 .1 is a graph illustrating silicon core width and totalwinding width vs. silicon core thicknesses for several differentelectroplating thicknesses and for 0.5μ thick tantalum. FIG. 2.28 .2 isa graph illustrating and silicon to silicon core spacing vs. siliconcore thicknesses for several different electroplating thicknesses andfor 0.5μ thick tantalum.

In general since one defines the frequency of operation, to maintain theR_(AC)/R_(DC)<2 condition, the encapsulating electro-plating metalthickness δ_(EP) is already a set parameter, leaving only one parametert_(3Si) for setting the HARMS process.

Since a good yielding and controllable HARMS process can be realizedwith some set rules, such as a minimum Ta deposition width of 2μ,thickness of 0.5μ and Si Core aspect ratio Δ_(SiW)<25 and the desiredelectro-plating metal thickness OEM, the relation between the desiredfinished dimensions of the spiral inductor and the Si Core dimensionswill be non-linear as shown in FIG. 2.29 -FIG. 2.30 . FIG. 2.29 showsthe needed Si Core thickness t_(3Si), to achieve the desired Δ_(w)values for δ_(EM)=2, 4, 6 and 8μ. As can be seen, as the δ_(EM)increases one needs a greater t_(3Si) to satisfy the “tight coupling”condition. The dashed line is the Δ_(w)=10 and the intersection pointbetween it and the family of curves for δ_(EM)=2, 4, 6 and 8μ will givethe needed t_(3Si) for the process to achieve this Δ_(w)=10 “tightcoupling” condition. As can be seen in FIG. 2.29 for most of theapplications, just from the processing point of the HARMS process,t_(3Si) has to be greater than 200μ! Another important relation is thecross-sectional area of the electro-plated metal area, which determinesthe resistance of the spiral inductor that is shown in FIG. 2.30 . Ascan be seen, by adjusting t_(3Si) and δ_(EM) one can achieve very largecross-sectional areas in a very small footprint, which cannot be donewith any prior art metallization rules and capabilities. The resultingmaximum electro-migration current-density limited current-carryingcapability and line resistance of the spiral windings for the length of5,000μ as a function of t_(3Si) and for δS_(EM)=2, 4, 6 and 8μ is shownin FIG. 2.31 .1 and FIG. 2.31 .2 for copper as the electro-platingmaterial in the calculations. In summary, all the needed goals of thetight coupling condition can be achieved with the inventive HARMSprocessing capabilities, along with extremely low line resistance andhigh current-carrying capability.

FIGS. 2.21-2.26 illustrate the results of processing a SiO wafer fortwo-sided connectivity. In FIG. 2.21 , a tantalum deposition is placedon the bottom or holding wafer with a region etched aligned to thespiral pad locations or STV locations, which are intended for connectionfrom the backside of the wafer. In FIG. 2.22 a Si Etch step results in adeep reactive ion etching to the buried oxide, SiO₂, as previouslyexplained. FIG. 2.23 is the result of a SiO etch step to remove thatlayer and expose the underside of the spiral pad or the bottom side ofthe STV. FIG. 2.24 shows the result of a further tantalum depositionstep coating the exposed walls and voids to establish electricalconnectivity across the whole pad or STV structure. FIG. 2.25illustrates the result of an electro-plating step covering the sameregion. Finally a C4 bump can be positioned at the void in electricalconnection with the pad or STV (FIG. 2.26 ). The whole chip andinterposer with this C4 bump (as part of an array of C4 bumps) are thenin condition to be mounted bump to bump.

The processing difficulty in having STV's using a SiO wafer is clearlydemonstrated in FIG. 2.21 -FIG. 2.26 . On the other hand, the benefitsof performance that one gets with Si Core thicknesses exceeding 200μ isalso evident. One does not need a Si Core thicknesses of less than 200μ.Having Si Core thickness larger than 200μ completely eliminates the needfor SiO wafers as starting materials and makes it possible produce aninterposer with very desirable feature of two-sided connectivity, andhaving STV's is a much easier process than shown in FIG. 2.21 to FIG.2.26 . Therefore processing becomes very simple by using just a regularSi wafer with a thickness of anything available in the range of 200μ andabove with a 1 to 2μ thick SiO₂ grown on one side of it as shown in FIG.2.32 is with bottom C4 solder bump.

Further Use and Application of the “HARMS” Interposer Technology: LowImpedance Power/Ground Delivery Network for High Power IC's

As can be seen in FIG. 2.31 , the metal rules in the “HARMS” (Thick AndHigh Aspect Ratio Metal) interposer technology can deliver very smallresistance values even compared to PCB technologies and with muchsmaller metal widths and spacing. This suggests that this discovery ofincreased capability is not limited to spiral inductor design. Theinvention can be extended to assisting in distributing power on the ICitself in order to simplify the IR drop issues in large IC's, which is aserious issue in high power processor designs.

Shrinking IC process geometries results in increasing current densitydistribution per area in IC's and lower supply voltages dictated byscaling rules. Distributing these high currents with an acceptablevoltage drop has always been a challenge, but now it is becoming an evena more difficult task. One of the main reasons of seeking FIVR is alsorelated to find a solution to this problem, dropping 1.8V supply to 1Vwith high efficiency on the IC itself which also can be controlled byprocessor activity.

“HARMS” interposer can give a fairly easy solution to this problem.Considering a t_(Si)=300μ and 8μ of Cu electro-plating giving totalwidth of 32μ of spacing width as shown in FIG. 2.28 one can strapVDD/VSS lines for long distances on the chip not previously achievableeven in PCB technologies having 2 ounce Cu which corresponds tothickness of 2.8 mil (71.12μ) and with widths of 10.94 mil (278μ)Moreover in addition to very small resistances VDD lines can be shieldedwith VSS lines, producing very significant drop in loop inductances.

The effect of Si Core thickness on the inductance of a simplepower/ground delivery network can be shown comparatively very clearlyfor cross-sections shown in FIG. 2.27 and FIG. 2.28 . Reference is madeto Table [8.1].

t = 50, w = 5, Δ_(w) = 10, l = 5 mm (5,000μ) R = 546 mΩ (L₁₁ = 5.703 nH,L₂₁ = L₁₂, L₂₃ = L₁₂, L₃₁ = L₁₃, L₃₂ = L₂₃) Half Loop S Inductance spaceL₁₂ L₁₃ L_(1/2loop)[nH] [μ] Δs = t/w [nH] [nH] (L₁₁-L₁₂) L₁₁/L_(1/2loop)C[pF] Z₀[Ω] f_(RES)[GHz] 25 2 4.656 4.078 1.048 5.442 0.3453 55.08 8.36810 5 5.107 4.656 0.5966 9.560 0.8633 26.29 7.013 5 10 5.301 4.938 0.402114.183 1.727 15.26 6.040 3.333 15 5.373 5.048 0.3304 17.260 2.59 11.295.441Table [8.1] illustrates that one can achieve over 3A current due toenlarged cross-sectional area with minimized resistance and inductance,whereas no one else today can come even close. The best that can be donetoday for 5 mm wire length (t=4μ, w=2.8μ, s=2.8μ and forJ_(MAX)=2×10⁶A/cm2) is a maximum current of less than 225 mA.L11=7.792nH, L12=6.467nH, L13=5.79InH, L_(looP=)1.325nH, R_(cu)=7.7Ω,S=11.2μ², IMAX=224mA. Conclusion: Width needed for the samecross-sectional area S=153μ²is 38.25μ

The columns 3-4 in Table [8.1] show the first row of the 3 by 3inductance matrix which corresponds to a VSS/VDD/VSS parallel powerdelivery network having a length of 5 mm (5,000μ) with a thickness of50μ and width of 5μ of Cu width. The even spacing between theVSS/VDD/VSS parallel power delivery network is 10μ and 5μ as shown inrows 2 and 3 in Table [8.1]. Column 6 is the half of the loop inductancefor just the VDD/VSS pair. Column 7 shows the ratio between the diagonalelement of the inductance matrix to the half loop inductance, which is ameasure of inductance reduction ratio compared to a loop having aninfinitely far return path. As can be seen, the reduction of theinductance is very significant. Such figures cannot even be achieved inPCB's with ground planes!

The note below Table [8.1] compares the prior art using the thickestpossible metal thickness of 4μ of Cu with space and width of 2.8μ.Inductance, resistance reduction and the increase of its currentcarrying capabilities of a power/ground delivery network using theinventive HARMS process, even with for a 50μ thick Si Core, as here,compared to the best suitable process for this application is simply offthe charts! The last note below the table shows the needed metal widthfor achieving the same resistance and current carrying capability usingthe thickest metal process being 38.25μ compared to w=5μ used in theHARMS process with basically insignificant reduction in the half loopinductance value compared to the given value of 1.325 nH where the HARMSprocess can give a value of 0.402 nH!

TABLE 8.2 t = 300, w = 30, Δ_(w) = 10, l = 5 mm (5,000μ) R = 15 mΩ (L₁₁= 3.924 nH L₂₁ = L₁₂, L₂₃ = L₁₂, L₃₁ = L₁₃, L₃₂ = L₁₂) Half Loop sInductance space L₁₂ L₁₃ L_(1/2loop)[nH] [μ] Δs = t/w [nH] [nH](L₁₁-L₁₂) L₁₁/L_(1/2loop) C[pF] Z₀[Ω] f_(RES)[GHz] 150 2 2.906 2.3541.018 3.855 0.3453 54.30 8.487 60 5 3.345 2.906 0.5791 6.776 0.863325.90 7.111 30 10 3.536 3.179 0.3882 10.108 1.727 14.99 6.148 20 153.606 3.287 0.3176 12.355 2.59 11.07 5.55 t = 300, w = 30, Δ_(w) = 10, l= 5 mm (5,000μ), R_(Cu) = 15.66 mΩ, S = 5,508μ², I_(MAX) = 110 A (L₂₁ =L₁₂, L₂₃ = L₁₂, L₃₁ = L₁₃, L₃₂ = L₁₂) The best which can be done todayfor 5 mm wire length (t = 4μ, w = 2.8μ, s = 2.8μ and for J_(MAX) = 2 ×10⁶ A/cm²) L₁₁ = 7.792 nH, L₁₂ = 6.467 nH, L₁₃ = 5.791 nH, L_(loop) =1.325 nH, R_(Cu) = 7.7 Ω, S = 11.2μ², I_(MAX) = 224 mA Width needed forthe same cross-sectional area S = 5,508μ² is 1,377μ (1.377 mm is largerthan a typical chip!)

Similarly columns 3-4 in Table [8.2] show the first row of the 3 by 3inductance matrix which corresponds to a VSS/VDD/VSS parallel powerdelivery network having a length of 5 mm (5,000μ) with a thickness of300μ of Cu width. The even spacing between the VSS/VDD/VSS parallelpower delivery network are taken as 60 and 30μ as shown in rows 2 and 3in Table [8.2]. The last note below Table [8.2] shows the needed metalwidth for achieving the same resistance and current carrying capabilityusing the thickest metal process being 1,377μ compared to w=30μ used inthe HARMS process with insignificant reduction in the half loopinductance value compared to the given value of 1.325 nH where the HARMSprocess can give a value of 0.388 nH!

Utilizing an interposer designed using the HARMS process as a VDD/VSSstrapping is very clearly demonstrated with the Table [8.1] and Table[8.2] is a great advantage in any high-current-consuming IC VDD/VSSnetwork design. Basically having the VSS shield in very close proximityto VDD, the very high mutual inductance obtained by this arrangementwill generate a very small supply loop inductance, much less than VDDline self-inductance by itself. In addition to this the power deliverynetwork will have a fairly large distributed capacitance with very smallseries resistance. The capacitance for 5 mm long VDD/VSS pair is givenas the 7^(th) column of Table [8.1] and Table [8.2] are significantlylarger values that without this invention are not possible to realize inany conventional IC processes today. None of these desirable propertiescan even be achievable in any packaging, MCM, thin film, thick film orany PCB technologies available today. Hence this invention represents asignificant advance.

Integrating Large Value Decoupling Capacitors Using PZT/PLZT Material asDielectric in Selected Areas in the HARMS Process

In order to guard against large current spikes in any IC there is a needfor placing high value decoupling capacitances as close as possible tothe noise generating circuits in the IC. Low impedance VDD/VSS networkdesign as explained above reduces this problem but due to high speed andsmall clock-skew requirements especially in processor designs, there isalways a need for high-value on-chip decoupling capacitors. Placinglarge-value on-chip decoupling capacitors wastes a significant area.However, these can be integrated in the HARMS process which already hasa very large distributed capacitance by depositing PZT/PLZT material asdielectric in selected areas between interdigitated fingers that formthe capacitance plates. These materials are also known as ferroelectricmaterials (Lead Zirconate Titanate) [Chemically PbZr_(x)Ti_((1-x))O₃(0≤x≤1)] and can have very large relative dielectric constants, on theorder of 300 to 10,000 depending to their doping, which can completelyeliminate the need for large value off-chip decoupling capacitors orlarge capacitor values needed in the circuit design. The capacitorstructure according to the invention is as shown in FIG. 2.35 . Twoelectrode each have vertical fingers disposed laterally adjacent in analternating or interdigitated configuration. While the application isprimarily for decoupling capacitors, other applications are alsocontemplated.

Integrated Controlled Impedance Transmission Lines in IC Technologies

Constructing controlled-impedance transmission lines in anysemiconductor process is a challenge. For semi-insulating substratesemiconductor technologies such as in GaAs IC's it is a simpler problembut due to the requirement of low resistivity lines they occupy a largespace. In silicon IC's, due to the finite conductivity of the siliconmaterial the issue of “slow-wave” phenomenon is a serious issue inaddition to the large area requirement.

The transmission line characteristic impedance Z₀ can be given as,

$\begin{matrix}{Z_{0} = \sqrt{\frac{R + {j\omega L}}{G + {j\omega C}}}} & (2.23)\end{matrix}$

where R, L, G, C and ω are resistance, inductance, conductance,capacitance per unit length and angular frequency respectively. As canbe seen in (2.23) one needs small R and G values compared to Lω and Cωto produce a good transmission line. This condition gives nearlyfrequency independent Z₀ with its well-known approximation as,

$\begin{matrix}{Z_{0} \cong {\sqrt{\frac{L}{C}}{for}R{\operatorname{<<}L}\omega{and}G{\operatorname{<<}C}{\omega.}}} & (2.24)\end{matrix}$

Since the Si Core technology disclosed herein gives very low resistanceper unit length values of legs or segments in a much smaller footprintas shown in FIG. 2.31 .2, one can build controlled characteristicimpedance transmission lines utilizing the invention. The parameters forbuilding transmission lines with desirable characteristic impedances onthe order of 50-75 ohms using much smaller area is displayed in Table[8.1] and Table [8.2]. The columns 8 of Table [8.1] and [8.2] shows thecharacteristic impedance values constructed as a pair ofhigh-aspect-ratio metallization elements as a function of spacing andtheir corresponding Δ_(S). A typical transmission line consists of threesegments, a center leg and two adjacent legs in the tight couplingcondition. This structure is not possible with any other known ICtechnology. The columns 9 in Table [8.1] and Table [8.2] are theresonance frequencies calculated lumped circuit assumption for 5 mm longtransmission lines.

Implementation of Spiral Structures in Flex Technology

In the absence of truly thick and high aspect ratio metal (HARMS)interposer technology, for test and verification purposes, spiralstructures were built and tested using Flex PCB technology according tothe present invention as hereinafter explained to access performanceimprovements and estimate performance gains when such an interposermetallization process became available. Flex PCB technology is wellknown as a flexible form of printed circuit board (PCB) technologywherein conductive layers insulated by a dielectric and formed flexiblesheets or strips are used to form circuitry. Flex PCB technology hasnever before been applied to the present use. Since development of anyIC metallization process with any non-standard requirements is anexpensive and a long process, it is helpful to show performance resultsusing a much cost effective way to justify the work. Nevertheless, itwas determined that Flex structures are also useful in and of themselvesand for this reason they are disclosed herein as a type of miniatureinductor. Two types of Flex PCB-based spiral structures are disclosedand have been constructed for test purposes and possibly practicalapplication, herein designated single-layer and folded single-layer,both forming stacked multiple-layer structures that come within thescope of the present invention.

The main purpose of the Flex structures disclosed herein is to be ableto generate structures that are easy to build with very limitedmechanical skill and resources and that verify theoretically derived“tight coupling” conditions. However, using miniature mechanicalproduction equipment, all of the Flex structures herein describe can beminiaturized with ease.

Type 1 Flex Structure: Flex Spiral Structures:

The first type of Flex PCB-based spiral structure was constructed byusing a standard Flex PCB technology with a number of different metalline widths wound around a dielectric core radius larger than theminimum radius of 32 mil (812.8μ) specified by the Flex technologyparameters shown in FIG. 3.1 . The standard technology parameters ofknown Flex PCB technology used are given in FIG. 3.1 , FIG. 3.2 and FIG.3.8 . Several inductors were built with different numbers of turns, andthe effects of width change, which corresponds to the “thickness” in theinterposer technology, were evaluated. As can be seen, although theseinductors are not as small as interposer inductors, these prototypeswere able to demonstrate the advantage of placing the wide dimensions ofthe conductors close together to produce inductance value andperformance improvement for small numbers of turns at low frequencies,thus experimentally demonstrating the advantage of the “tight couplingcondition.”. This outcome also demonstrates the ease of design of theclaimed invention in practice and an was an easy and cost-effective wayof experimental confirmation of the validity of the assumptions made intheir theoretical derivation.

The thickness of rolled copper according to the standard Flex PCBtechnology process employed is shown in FIG. 3.1 as 1.4 mil (35.56μ).This corresponds to skin depth δ dependent desirable widths todemonstrate the usefulness of the invention in the 25-50 MHz range ofapplications as shown in Table [2]. As an alternative, gold (Au) can beemployed as the conductive material, which permits much thinnerstructures with a much smaller radius of curvature. Since one cannotchange the copper thickness, the high aspect ratio requirements can bedemonstrated by making the tracks wider, which can be easily donewithout violating any standard Flex technology. Having a metal-widthaspect-ratio goal of Δ_(M)=10, one needs a minimum track width of 14mil. On the other hand to achieve the adjacent metal spacing aspectratio of Δ_(S)=5, using the numbers given in FIG. 3.8 one needs 2.6×5=13mil of track width. This Δ_(S)=5 figure can be extended to Δ_(S)=10 byusing a track width of 2.6×10=26 mil. Therefore building a Flex spiralwith a 14 mil (355.6μ)-wide Flex track demonstrates the theoreticalderivation of the invention, which can be very easily realized. If allthe assumptions made are correct, to experimentally verify thederivations, one needs to build Flex spiral structures having 14 miltracks as well as Flex spiral track widths wider and narrower than 14mil, then compare their performances with measurements. Starting fromthe minimum track width in Flex technology of 3 mil (76.2μ), sixdifferent Flex spiral structures built with track widths of 3, 6, 12,13, 15 and 26 mil will demonstrate the full range of simulatedstructures.

A major issue to consider is the connection of Flex spiral structures toa PCB, where several arrangements have to be made for measurements. Oneobvious choice is to use a Flex-to-PCB connector, such as a Type52015-3TE AMP. This 3-pin flat connector has a width of 493 mil(12,522.2μ) and a length of 273 mil (6,934.2μ). With connectiondimensions as shown in FIG. 3.3 the Flex-to-PCB connector extensionregion will not be less than 500 mil (12,700μ). Therefore the length ofthe Flex spiral structure must to be on the order of 5 to 10 times thislength to provide good electrical measurements, giving a length of 6.35to 12.7 cm! As an example if a Flex length of d_(x)=500 mil (12,700μ or12.7 cm) is used, the geometrical length of the spiral inductor becomesapproximately 10 times the connector length, and the trace lengthsassociated with the measurement circuit/fixture combined and theelectrical measurements will be dominated by the spiral inductor ratherthan by parasitics introduced by the external circuit.

Since the connection to a PCB disposed below will be done as shown inFIG. 3.7 , the inner dimension d_(IN) of the Flex spiral cannot be lessthan the width of the Flex trace in the connector extension region.Leaving 10 mil of spacing from both edges of the Flex ribbon, themanageable minimum inner dimension becomes 408 mil (10,363.2μ). For someease in manual winding and handling, d_(IN)=500 mil is used as shown inFIG. 3.6 .

Having d_(IN)=500 mil and the trying to meet the goal of achieving aspiral-inductor-dominated measurements, the minimum number of turnsshould be greater than five. A five-turn Flex spiral inductor will havespiral winding length of 9,500 mil (plus the needed extension sizes forconnections as described above).

Using a three-pin connector, such as a Type 52015-3TE AMP connector,allows embedding three different trace widths for the same number ofturns in the same Flex layout and brings space saving advantage in ameasurement setup. Vertically stacked three-inductor Flex structures asshown in FIG. 3.4 and FIG. 3.5 also allow coupling electricalmeasurements to be made and compared to a single trace spiral inductorperformance as shown in FIG. 3.3 .

The Flex spiral with three traces having 5, 10, 20 trace widths with10-mil spacing and 10-mil clearance from the edges as shown in FIG. 3.5will give a Flex spiral height of 75 mil (1,905μ, 1.905 mm) on top ofthe PCB as shown in FIG. 3.7 . As a result, overall dimensions of thethree vertically stacked flex inductors then become approximately 540mil by 540 mil (13,716μ or 13.716 mm) with a height of 1,905μ or 1.905mm, which is a manageable handling dimension.

The end points of the Flex-to-PCB connector extension region havestiffeners to assure good connectivity between the Flex traces and theconnector. The PCB cutouts should be at least 500 mil high and wideenough for the Flex-plus-stiffener thickness to allow the Flexconnection to connect to the underlying PCB. Having an underpass regionat a distance of the PCB thickness away from the bottom of the Flexspiral for its B connection reduces the capacitive and inductivecoupling to the windings, which also improves performance.

Type 1 Flex Spiral Structure for PCB Transformer and Balun Structureswith A SINGLE NUMBER of Mechanical Winding Process for DesiredVoltage/Current/Impedance Conversion Ratios

Many applications require small, high-performance custom transformersand baluns on a PCB. FIG. 3.9 .1-FIG. 3.11 show how simple and easythese structures can be built using two-layer Flex technology by justadjusting the length and width of the second layer flex metal comparedto the first layer metal. The voltage division ratio is simply theratios of the lengths of first and second layer metal. As it is known,in any kind of a transformer, the turn ratio between the primary andsecondary windings determines the voltage ratio between the primary andsecondary windings. In the prior art way of building transformer/balunstructures, one has to go through a mechanical winding process equal tothe number of isolated windings, which is difficult and time consuming.In the making of the structures shown in FIG. 3.9 -FIG. 3.11 there isonly one mechanical winding step. The equivalent number of turn for eachisolated inductor (n₁, n₂, n₃, . . . ) will be automatically determinedby their length ratios (d₁, d₂, d₃, . . . ) making it very easy todetermine and build as a single Flex structure. As can be seen, anynumber of isolated windings can be made with any desired voltageconversion ratio between them with only two-layer Flex technology,providing the length is sufficiently large. This technique can beextended for large high power transformers as well.

The center area of such a wound structure can also have a magnetic core,which is another way of increasing the inductor values verysignificantly in the same area for miniature transformer/balun/inductorstructures very easily.

Type 2 Flex Structure: Folded Flex Coil

The second type of Flex-based spiral structure, the folded Flex coil,also demonstrate the multi-layer stacked inductor structures withdimensions similar to type 1 Flex inductors. Since high via resistanceand current redistribution in the via-to-trace transition region aremajor factors in performance limitation and area of a PCB, the inventivemulti-layer Flex inductors may be built with no vias by folding aserpentine structure on itself. This way one can have a single layerflex structure with a serpentine pattern as shown in FIG. 4.1 . The Flexribbon is fan folded in alternating directions along parallel axes, asindicated by dotted lines in FIG. 4.1 . The fan fold is done as manytimes as needed to generate as many turns desired. The number ofalternating folding operations is twice the number of turns desired.This is a cost- and area-efficient way of making stacked multi-layerinductors using only a single layer flex technology.

The serpentine metallization track widths may be the same used in a Type1 Flex structure, being 3, 6, 12, 13, 15 or 20 mil, drawn around thesame inner space d_(IN)=500 mil to make a fair experimental comparisonbetween vertical and horizontal placement of the windings with samewidth and spacing. Such structures were built and tested. To makecompatible inductor geometries with the Type 1 Flex Spiral structures,the Flex coil structures had l=500 mil and ƒ=100 mil, which is a numberlarger than the twice the minimum radius of curvature (32 mil) allowed.As an alternative, gold (Au) can be employed as the conductive material,which permits much thinner structures with a much smaller radius ofcurvature.

A Flex track width of 13 mil and 10 mil of extension from the trackedges gives a Flex ribbon width of 546 mil. For 5-turn coils, 10alternating folds of the periodic structure gives 6,130 mil of length,which is less than the Flex length of 9,500 mil as calculated in theType 1 Flex spiral structure. The total thickness of a 5-turn coil isthus about 40 mil.

This fan folding technique can be used to generate any spiral geometrywhich has a point symmetry property in the center of the fold line notedas P_(SYM) in FIG. 4.3 , including curves or other structures that arenot straight between folds. Common desirable structures are circles andoctagons along square shaped inner areas as shown in FIG. 4.3 . Thesehalf-circular coil structures resemble the coils built by advancedpackaging techniques shown in the FIVR work, having no vias which doesnot perturb the current flow and gives better Q and lower seriesresistance associated with the prior art coils. [3-5]

Comparing the manufacturability and performance of Type 1 and Type 2Flex inductors the selection of a Type 1 Flex spiral is a better choice.The most important use of a Type 2 Flex structure was in the fairexperimental comparison of very similar two different metal windingstructures. The much better Q performance of a vertically placedhigh-aspect-ratio arrangement as in the HARMS process of constructingwindings, compared to the stacked horizontal placements of windings,both having the same width, spacing and same aspect ratios giving thesame inductance matrices, shows that the assumption made in the ACcurrent density distribution is valid and gives much more uniform ACcurrent distribution at high frequencies with significantly lower ACresistance. One naturally thinks that AC current density distributionand resulting AC resistance of a rectangular cross-sectional wire shouldbe independent of its orientation in space. This assumption is entirelytrue for a free standing straight wire in space, but it is not true whenit is formed as a spiral or a coil structure! It will be proper to sayMaxwell's equations, Ampere's law and the Helmholtz wave equation is atwork for this very practical and useful conclusion!

With the development of High Aspect Ratio Metallization (HARMS)interposer processes, there is a path for increasing the effectiveinductance of a spiral inductor, herein the “Tight Coupling Condition.”In a specific embodiment, four adjacent windings made according to theprocess couple more tightly, increasing the spiral inductance with theresult of increasing the Q of the spiral inductance at all frequencies.This configuration corresponds to the Type 2 Flex structure and to thesecond of the aspect ratio parameters for metallization.

The interposer process is not only a thick metal process that is muchthicker than is commercially available, it requires a high-aspect-ratiospacing rule as well. This is not the same as the known “thick metalprocess,” wherein the thickest metal evidently available today in ICprocessing is a mere 4μ thickness with 2.8μ width and 2.8μ spacing.Moreover, according to the calculations in support of the presentinvention, the width of the metal must be on the order of skin depth,but thicker than 50μ in most of the cases, to meet the desired DCresistance and electro-migration rules. If the metallization rules candeliver these metal width/thickness dimensions with only large spacings,more than the metal thickness, then a device will never be able toachieve the Tight Coupling Condition.

The invention has now been explained with reference to specificembodiments. Other embodiments will be evident to those of skill in theart. Therefore, it is not intended that this invention be limited tothose specific embodiments. Rather the invention is to be defined by theappended claims.

1.-25. (canceled)
 26. A miniature transmission line comprising: asemiconductor substrate constraining a maximum size of the transmissionline; a plurality of metal conductive legs formed in the semiconductorsubstrate, said metal conductive legs disposed with spacingtherebetween; the legs being configured with a high aspect ratio definedas a thickness dimension much greater than a width dimension and whereinlegs are closely spaced on a scale comparable to the width dimension toattain a tight coupling condition defined as a high coupling coefficientacross multiple legs.
 27. The transmission line of claim 26, the legshaving a thickness-to-width ratio of at least ten and a width-to-spacingratio between adjacent legs of at least five such that the tightcoupling condition is attained with a coupling coefficient of at least0.5 at the second most adjacent leg.
 28. The transmission line of claim26 wherein the semiconductor substrate is an interposer, furtherincluding terminals of the legs and pads coupled to the legs through theterminals to provide for external electrical connections. 29.-32.(canceled)